Power Play | CBSE Class 8 Maths Notes

This note covers repeated multiplication, exponential notation, laws of exponents, zero and negative powers, counting combinations, powers of ten, scientific notation, estimates, linear and exponential growth, and names for very large numbers.

How does folding paper reveal exponential growth?

Imagine a sheet of paper whose initial thickness is 0.001 cm, where cm means centimetre. Suppose it can be folded as many times as required. Each fold doubles the thickness, so the new thickness is twice the previous thickness.

This is a thought experiment: the calculation assumes that repeated folding remains possible. It explores what doubling does to a quantity. It does not establish a practical method of folding an ordinary sheet all the way to the Moon.

What happens after successive folds?

The symbol × means multiplication, = means equality, and ≈ means approximately equal to. The table gives selected thicknesses. An approximate value is a value close to the quantity being represented, rather than an exact statement of it.

After ten folds the thickness is just above one centimetre. After 26 folds it is approximately 670 m, where m means metre. After 30 folds it is about 10.7 km, where km means kilometre. After 46 folds it exceeds 7,00,000 km.

Definition: Exponential growth, also called multiplicative growth, changes a quantity by repeated multiplication. In the paper experiment, each fold multiplies the thickness by two.

Compare equal groups of folds. Any three further folds multiply the existing thickness by 2 × 2 × 2, which is eight. Any ten further folds multiply it by 1024. This multiplier, the number by which the thickness is multiplied, depends on the number of further folds.

Worked example 1. A sheet starts at 0.001 cm and doubles in thickness at every fold. Find its thickness after three folds.

Answer: Multiply by two three times: 0.001 × 2 × 2 × 2 = 0.008 cm. Count one multiplication for each fold.

What do a base and an exponent mean?

Exponential notation is a compact way to write repeated multiplication. In 5⁴ = 5 × 5 × 5 × 5 = 625, the base is five, the repeated factor. A factor is a number being multiplied. The exponent is four, the number of copies of that factor.

Definition: For a counting number a, the expression nᵃ means a copies of the number n multiplied together. Here n is the base and a is the exponent. Counting numbers are 1, 2, 3 and so on.

A letter standing for a number is a letter-number. The expressions n² and n³ are read as n squared and n cubed. A square is a number multiplied by itself; a cube uses three copies of the number.

How are addition and multiplication different?

The symbol + means addition. Thus 4 + 4 + 4 = 3 × 4 = 12, whereas 4 × 4 × 4 = 4³ = 64. Count the repeated factors when forming a power; do not merely multiply the base by the exponent.

The sign − denotes a negative number or subtraction. Parentheses group an expression. In (−4)³, the entire number negative four is the base. Its three factors give −64. In (−2)⁴, four negative factors give the positive value 16.

How can prime factors be grouped?

Whole numbers are zero and the counting numbers. A prime number is a whole number greater than one with exactly two positive factors, one and itself. A positive factor divides the number exactly. Prime factorisation writes a number as a product, the result of multiplication, of prime numbers.

Worked example 2. Express 32400 using powers of its prime factors.

Answer: 32400 = 2 × 2 × 2 × 2 × 5 × 5 × 3 × 3 × 3 × 3 = 2⁴ × 5² × 3⁴. Each exponent counts the occurrences of its own prime factor.

How do you multiply powers with the same base?

Property: Add exponents when the base is the same

Let n be a base and let a and b be counting-number exponents. Then nᵃ × nᵇ = nᵃ⁺ᵇ. The exponent a + b counts all the factors after the two groups are joined. The base remains n throughout.

This property follows directly from repeated multiplication. The first power contains a copies of n and the second contains b copies. Their product, meaning the result of multiplication, contains both groups. Nothing in this operation changes the value of the individual repeated factor.

Worked example 3. Find 3⁷ using 3⁴ = 81 and 3³ = 27.

Answer: Split seven factors into a group of four and a group of three. Then 3⁷ = 3⁴ × 3³ = 81 × 27 = 2187.

How does the diamond puzzle use this idea?

A king has three daughters. Each receives three baskets; each basket contains three keys; each key opens three rooms. Each room has three tables, each table has three necklaces, and each necklace has three diamonds. Every stage multiplies the previous count by three.

There are four factors of three from the daughters through the rooms, giving 3⁴ rooms. Continuing through tables, necklaces and diamonds gives seven factors altogether. Therefore the number of diamonds is 3⁷, or 2187. Grouping factors avoids repeating the whole calculation from the start.

The label King appears above branches to three pictured daughters. Beneath each daughter are three baskets, with further branches to keys and rooms. The labels identify daughters, baskets, keys and rooms.

Reference: NCERT Class 8, page 23, unnumbered diagram

The same reasoning works with a letter-number. If p represents a number, p⁴ × p⁶ contains ten factors equal to p and becomes p¹⁰. Adding exponents is justified because both powers have the same base.

How do powers of powers and equal exponents work?

Property: Multiply exponents in a power of a power

For counting numbers a and b, (nᵃ)ᵇ = nᵃᵇ, where ab means a × b. The outer exponent counts groups; the inner exponent counts factors within each group. Multiplying them gives the total number of repeated factors.

Worked example 4. Evaluate 4⁶ by grouping its factors in two ways.

Answer: 4⁶ = (4³)² = 64 × 64 = 4096. Alternatively, 4⁶ = (4²)³ = 16 × 16 × 16 = 4096. Both arrangements contain six factors of four.

This also explains why (2²)⁵ and (2⁵)² both equal 2¹⁰. The groups differ in size and number, but each arrangement uses ten factors of two. Do not add the inner and outer exponents in this operation.

Property: Combine bases when exponents are equal

Let m and n represent numbers and a a counting number. Then mᵃ × nᵃ = (mn)ᵃ, where mn means m × n. Pair each factor from the first power with one from the second power, keeping the common exponent.

Damayanti places one lotus in a pond where the number doubles daily. After four days she transfers all the lotuses to a pond where the number triples daily. After four more days, the count is 2⁴ × 3⁴ = 6⁴. Reversing the pond order gives the same count.

The symbol ÷ and a slash denote division. The numerator is the number above a fraction bar and the denominator is the number below it. With a non-zero denominator, the equal-exponent division rule is mᵃ ÷ nᵃ = (m ÷ n)ᵃ.

Worked example 5. Simplify 10⁴ ÷ 5⁴.

Answer: Pair the four numerator factors with the four denominator factors: 10⁴ ÷ 5⁴ = (10 ÷ 5)⁴ = 2⁴ = 16.

How do multiplication and powers count combinations?

A combination here means a possible selection or arrangement satisfying the stated choices. Estu has four dresses and three caps. For each cap, all four dresses are possible, giving 4 × 3 = 12 dress-and-cap combinations.

Counting dress first gives the same result: each of four dresses can be paired with three caps. This gives 3 + 3 + 3 + 3, again twelve. The calculation counts each complete outfit once rather than adding the numbers of separate items.

How does a shorter password help?

A difficult counting problem can become clearer by first solving a simpler version. For a two-digit password, each position has ten choices, from zero to nine. Zero may be the first digit and digits may repeat, so 00 is included.

1. There are ten choices for the first position.
2. Each first choice allows ten choices for the second position, giving 10 × 10 = 100 passwords.
3. Each two-digit arrangement allows ten third digits, producing 100 × 10 = 1000 three-digit passwords.
4. Continue the same reasoning until every position has been included.

Worked example 6. Count five-digit passwords when each position accepts any digit from zero to nine, repetitions and initial zeroes included.

Answer: 10 × 10 × 10 × 10 × 10 = 10⁵ = 1,00,000 passwords. The permitted arrangements run from 00000 to 99999.

A six-slot lock using the letters A to Z has 26 choices at each slot. Under the same repetition rule, its count is 26⁶. The exponent counts positions, while the base counts choices available independently at each position.

Read the conditions before counting. A password may start with zero even though an ordinary five-digit number does not. The multiplication used here depends on every position retaining all the stated choices.

Why do division and zero exponents follow the same pattern?

A quotient is the result of division. Consider a line of length sixteen units, written as 2⁴ units. Halving means dividing by two. Halving once gives 2³ units, twice gives 2² units, and three times gives 2¹ units.

Property: Subtract exponents when dividing the same base

Initially let a and b be counting numbers with a greater than b. For a non-zero base n, nᵃ ÷ nᵇ = nᵃ⁻ᵇ. Dividing removes matching factors from the numerator and denominator, leaving a − b copies of the base.

Worked example 7. Express 2¹⁰⁰ ÷ 2²⁵ as a power of two.

Answer: Both powers have the same non-zero base, so subtract their exponents: 2¹⁰⁰ ÷ 2²⁵ = 2⁷⁵. There is no need to calculate either large power first.

The condition on the base matters. If the base were zero, division would involve zero in the denominator. Such division is undefined, meaning that it has no assigned numerical value in these operations.

Property: A non-zero base raised to zero equals one

Dividing a non-zero quantity by itself gives one. At the same time, subtracting equal exponents gives zero. Consequently, n⁰ = 1 for n ≠ 0, where ≠ means “is not equal to”.

For example, 2⁴ ÷ 2⁴ = 1, while the exponent calculation gives 2⁰. Defining 2⁰ as one keeps the pattern consistent. The zero exponent concerns the exponent, not the value of the base.

Note: The statement n⁰ = 1 requires n to be non-zero. It must not be used to assign a value to 0⁰. Positive powers of zero and zero powers of non-zero numbers are different cases.

What do negative exponents and power lines show?

Continue halving the sixteen-unit line beyond four halvings. After five halvings, its length is 2⁴ ÷ 2⁵ = 2⁻¹ = 1/2 unit. After ten halvings, it is 2⁴ ÷ 2¹⁰ = 2⁻⁶ = 1/64 unit.

These expressions extend the exponent pattern below zero. Integers include the counting numbers, zero and the negatives of counting numbers. Negative exponents allow the laws of powers to work for integer exponents, provided the quantities involved are defined.

Property: A negative exponent gives a reciprocal

The reciprocal of a non-zero number is one divided by that number. For non-zero n, n⁻ᵃ = 1/nᵃ. In particular, 10⁻³ = 1/10³ and 7⁻² = 1/7². A negative exponent does not itself make the resulting number negative.

Worked example 8. Simplify 2⁻⁴ × 2⁷.

Answer: Add the exponents because the base is the same: 2⁻⁴ × 2⁷ = 2³. The sum of negative four and seven is three, so the product is a positive power of two.

To apply several exponent laws, identify the operation first. Multiplication of like bases adds exponents; division subtracts them; a power raised to another power multiplies them. The sign of an exponent does not change which operation is required.

How can a power line organise calculations?

A vertical line pairs powers from 4⁸ down to 4⁻² with their values. The labels include 4⁰ beside 1, 4⁻¹ beside 1/4, and 4⁻² beside 1/16. Curved arrows show multiplication and division by four or sixteen.

Reference: NCERT Class 8, page 29, unnumbered diagram

The line links neighbouring powers through multiplication or division by four. Comparing powers is also a division problem: 4⁷ ÷ 4⁵ = 4². Thus 16384 is sixteen times 1024. A comparison asks for the multiplying factor between quantities, not their difference.

How do powers of ten explain place value?

Place value is the contribution of a digit determined by its position in a number. Moving through units, tens, hundreds and thousands gives powers of ten. Each place to the left is worth ten times the neighbouring place to its right.

How is a whole number expanded?

An expanded form writes a number as a sum of its place-value contributions. For example, 47561 = (4 × 10000) + (7 × 1000) + (5 × 100) + (6 × 10) + 1.

Using powers, the same number is (4 × 10⁴) + (7 × 10³) + (5 × 10²) + (6 × 10¹) + (1 × 10⁰). The exponent falls by one as each successive digit is read from left to right.

The units contribution uses 10⁰ because 10⁰ equals one. Writing that term explicitly helps connect whole-number places to decimal places. No separate rule is needed when the expansion reaches the decimal point.

How do negative powers describe decimal places?

The first place after the decimal point represents tenths, or 10⁻¹. The next represents hundredths, or 10⁻², followed by thousandths, or 10⁻³. Each is one tenth of the place immediately to its left.

Worked example 9. Write 561.903 in expanded form using powers of ten.

Answer: 561.903 = (5 × 10²) + (6 × 10¹) + (1 × 10⁰) + (9 × 10⁻¹) + (0 × 10⁻²) + (3 × 10⁻³).

The zero in the hundredths place contributes zero, but its position still matters. It keeps the three in the thousandths place. Reading the digits and their positions together prevents confusing a decimal number with another number containing the same non-zero digits.

This expansion connects positive, zero and negative exponents in one system. Positive exponents describe places above units, zero describes units, and negative exponents describe places smaller than one.

How is scientific notation written and compared?

Scientific notation, also called standard form, expresses a positive number as x × 10ʸ. Here x is the coefficient, the number multiplying the power of ten, and y is an integer exponent.

The coefficient satisfies x ≥ 1 and x < 10, where ≥ means greater than or equal to and < means less than. Several exponential expressions may represent the same number, but this coefficient condition selects its standard form.

How can equivalent expressions be distinguished?

The number 5900 can be written as 590 × 10¹, 59 × 10², 5.9 × 10³ or 0.59 × 10⁴. All have the same value. Of these, 5.9 × 10³ meets the coefficient condition.

Worked example 10. Express 59,853 in standard form.

Answer: 59,853 = 5.9853 × 10⁴. The coefficient 5.9853 is at least one and less than ten. Multiplying it by 10⁴ restores the original number.

When comparing positive numbers already in standard form, first compare their exponents. If the exponents are equal, compare their coefficients. Checking the coefficient condition first prevents an incorrect comparison of expressions written in different forms.

How does this help compare distances?

These distances are rounded-off estimates, averages or approximations. The Sun-to-Earth distance is the smallest: its standard-form exponent is eleven rather than twelve. Of the other two, the Saturn-to-Uranus distance is greater because 1.439 is greater than 1.4335.

Often, the exponent is more important than the coefficient. It identifies the scale of the number. Comparing coefficients alone would miss the effect of the different powers of ten.

How do estimation and growth models help solve problems?

An estimate is an approximate value used when an exact answer is unavailable or unnecessary. An assumption is a value or condition adopted for a calculation. A mathematical model describes relationships between quantities so that the problem can be calculated.

What steps make an estimate useful?

1. Make a quick guess before performing calculations.
2. Identify the quantities involved and express the relationship between them.
3. State reasonable assumptions or approximations for unknown information.
4. Calculate, then compare the result with the original guess.

Different assumptions can give different answers. The aim is a reasonably close estimate supported by a suitable relationship. Extra digits should not suggest greater certainty than the information provides. In most cases involving large numbers, the size of the quantity matters more than its exact value.

Worked example 11. Assume Roxie's weight is 45 kg, where kg means kilogram, and jaggery costs ₹70 per kilogram, where ₹ denotes rupees. Find the value of jaggery equal to her weight.

Answer: Value = weight × price per kilogram = 45 × 70 = ₹3150. This result depends on the stated weight and price assumptions.

How does additive growth differ from multiplicative growth?

Linear growth adds a fixed amount at each step. In an imagined ladder to the Moon, assume each step gains 20 cm and the distance is 3,84,400 km. Each additional step increases the height by the same twenty centimetres.

The calculation gives 1,92,20,00,000 steps, or 192 crore and 20 lakh steps, where a crore is 10⁷ and a lakh is 10⁵. This is additive growth.

In the folding experiment, each fold instead doubles the thickness; 46 folds would take the thickness beyond the Moon's distance under the model's assumptions.

The difference is between adding the same amount and multiplying by the same factor. For paper, the amount added becomes larger as the thickness grows. For the ladder, the height gained per step remains twenty centimetres.

Precision, the detail conveyed by the digits reported, also matters when giving estimates. If a population is known only to around 1 lakh 42 thousand, write 1.42 × 10⁵. If it is known only to around 1 lakh 40 thousand, write 1.4 × 10⁵.

How can powers of ten make very large quantities meaningful?

Large quantities become easier to interpret when related to familiar names, times or other quantities. A lakh is 10⁵ and a crore is 10⁷. An arab is 10⁹, the same as an international billion. An international million is 10⁶.

How do comparisons use scientific notation?

The estimated global ant population is 2 × 10¹⁶. Taking the human population as about 8 × 10⁹ makes it possible to estimate ants per person. “Per person” means dividing the total number of ants by the number of people.

Worked example 12. Estimate ants per human using about 2 × 10¹⁶ ants and about 8 × 10⁹ humans.

Answer: Divide coefficients and powers: (2 ÷ 8) × 10¹⁶⁻⁹ = 0.25 × 10⁷ = 2.5 × 10⁶. There are about 2.5 million ants per human using these estimates.

The conversion to 2.5 × 10⁶ keeps the coefficient in standard form. Retaining “about” is essential: calculating with estimated populations does not produce an exact population ratio. The same principle applies to calculations with estimated numbers of trees, birds or stars.

What do large numbers of seconds feel like?

A million seconds is less than a fortnight, meaning two weeks, whereas a billion seconds is approximately 31.7 years. The three extra powers of ten create a very large change in duration. Reading the exponent helps reveal this change more clearly than counting zeroes.

Relating unfamiliar quantities to familiar durations builds a sense of their size. The approximate comparisons remain useful without pretending that the rounded durations in the table are exact conversions.

How have people named and represented enormous numbers?

Names for large numbers can themselves follow multiplication patterns. In the Indian system, one hundred thousand makes a lakh, one hundred lakhs makes a crore, one hundred crores makes an arab, and one hundred arab makes a kharab, or 10¹¹.

Which patterns continue beyond a billion?

One hundred kharab makes a neel, or 10¹³. One hundred neel makes a padma, or 10¹⁵. One hundred padma makes a shankh, or 10¹⁷, and one hundred shankh makes a maha shankh, or 10¹⁹.

In the American/International system, one thousand millions makes a billion, one thousand billions makes a trillion, or 10¹², and one thousand trillions makes a quadrillion, or 10¹⁵. Compare powers of ten when relating the two naming systems.

What do historical examples show?

The Lalitavistara, a Buddhist treatise from the first century BCE, includes names for odd powers of ten up to 10⁵³. BCE means Before Common Era. A treatise is a written work discussing a subject in detail.

Mahaviracharya's Ganita-sara-sangraha lists twenty-four terms, extending to 10²³. The anonymous Jaina work Amalasiddhi gives a name for each power of ten up to 10⁹⁶. These examples show that naming extremely large numbers has a long history.

A googol is 10¹⁰⁰. A googolplex is ten raised to the power of a googol. In the second expression, the exponent is itself an enormous number. Distinguishing a large base from a large exponent is therefore essential.

Exponential notation makes such numbers concise to write even when their size is difficult to imagine. The same notation used for folded paper and passwords can describe quantities far beyond everyday experience, while the laws of exponents provide a consistent way to work with them.

Glossary

• Exponential notation — A compact representation of repeated multiplication, using a base and an exponent to record repeated factors.
• Base — The number or letter-number whose repeated multiplication is represented by a power.
• Exponent — For a positive integer power, the number of copies of the base multiplied together.
• Prime factorisation — Writing a whole number greater than one as a product of prime numbers.
• Reciprocal — One divided by a non-zero number; negative powers connect a power with its reciprocal.
• Integer — A number that is zero, a counting number, or the negative of a counting number.
• Place value — The contribution of a digit determined by its position within the written number.
• Scientific notation — Writing a positive number as a coefficient at least one and less than ten, multiplied by an integer power of ten.
• Coefficient — The numerical multiplier placed before the power of ten in scientific notation.
• Estimate — An approximate value useful when a quantity is not known exactly or an exact answer is unnecessary.
• Linear growth — Growth produced by adding a fixed amount at each successive step of a process.
• Exponential growth — Multiplicative growth in which a quantity is repeatedly multiplied by a fixed factor.

Common errors and misconceptions

• Misconception: 4³ means three times four. Correct: It means 4 × 4 × 4 = 64. Repeated addition and repeated multiplication are different operations.
• Misconception: Multiply the exponents whenever two powers are multiplied. Correct: With the same base, add exponents. Multiply exponents when raising a power to another power.
• Misconception: A zero exponent makes the answer zero. Correct: A non-zero base raised to zero equals one. The requirement that the base is non-zero must be retained.
• Misconception: A negative exponent produces a negative answer. Correct: It gives a reciprocal: 2⁻¹ = 1/2. The negative sign in the exponent does not make the base negative.
• Misconception: Any coefficient is acceptable in standard form. Correct: It must be at least one and less than ten. For 5900, use 5.9 × 10³.
• Misconception: A five-digit password cannot start with zero. Correct: The password model allows initial zeroes and repeated digits, so it includes arrangements such as 00000.
• Misconception: A calculation using estimates produces an exact real-world answer. Correct: The result remains dependent on the approximate inputs and assumptions; preserve words such as “about”.

Exam-style questions with model answers

Q1. In 5⁴ = 625, identify the base and exponent and explain what each represents. [2 marks]

1. The base is five. It is the number repeated as a factor in the multiplication.
2. The exponent is four. It tells us to multiply four copies of five: 5 × 5 × 5 × 5 = 625.

Q2. A sheet is initially 0.001 cm thick. Assuming each fold doubles its thickness, find the thickness after three folds and explain the operation. [3 marks]

1. Each fold multiplies the preceding thickness by two, so three folds require three successive multiplications by two.
2. The thickness is therefore represented by 0.001 × 2³ cm, where the exponent three counts the folds.
3. Since 2³ = 8, the thickness is 0.001 × 8 = 0.008 cm under the stated folding assumption.

Q3. Explain and evaluate 2⁴ ÷ 2⁴ and 2⁴ ÷ 2⁵ using exponent laws. [4 marks]

1. For division with the same non-zero base, subtract the denominator's exponent from the numerator's exponent.
2. Thus 2⁴ ÷ 2⁴ = 2⁰ = 1, since a non-zero number divided by itself equals one.
3. For the second quotient, 2⁴ ÷ 2⁵ = 2⁻¹ because four minus five is negative one.
4. A negative exponent gives a reciprocal, so 2⁻¹ = 1/2. These calculations explain both the zero power and the negative power.

Q4. A password has five positions. Each accepts any digit from zero to nine, including repeated digits and an initial zero. How many passwords are possible? [3 marks]

1. Each position has ten choices. Allowing repetitions means that choosing one digit does not remove it from later positions.
2. Multiply the choices for all five positions: 10 × 10 × 10 × 10 × 10 = 10⁵.
3. Therefore there are 1,00,000 possible passwords. Initial zeroes are permitted, so arrangements beginning with zero are included in this count.

Q5. A dairy needs unique identifier codes for 8.5 billion milk packets, with one code per packet. Each code has the same length and uses digits zero to nine. Repetitions and initial zeroes are allowed. Find the minimum code length and justify it. [5 marks]

1. One billion is 10⁹, so the required number of distinct codes is 8.5 × 10⁹, or 8,500,000,000 codes.
2. Let n be the number of positions in each code. Every position has ten choices, giving 10ⁿ possible codes.
3. Nine positions give 10⁹ codes. This is fewer than 8.5 × 10⁹, so nine positions cannot cover all the packets.
4. Ten positions give 10¹⁰ codes. This is more than 8.5 × 10⁹, so ten positions provide enough distinct arrangements.
5. The minimum length is therefore ten digits. Each extra position multiplies the available count by ten, and the preceding length is insufficient.

Q6. Use an estimated 2 × 10¹⁶ ants and about 8 × 10⁹ humans to estimate ants per human. Give the answer in scientific notation and explain its precision. [5 marks]

1. The required relationship is total ants divided by total humans, because the question asks for the number of ants per person.
2. Divide the coefficients: two divided by eight is 0.25. This gives the numerical multiplier before the powers are simplified.
3. Divide the powers of ten by subtracting exponents: 10¹⁶ ÷ 10⁹ = 10⁷. The estimate is therefore 0.25 × 10⁷.
4. Rewrite this as 2.5 × 10⁶, placing the coefficient between one and ten as required for scientific notation.
5. There are about 2.5 million ants per human using the supplied values. The answer is approximate because both population inputs are estimates.

Q7. Assume Roxie weighs 45 kg and jaggery costs ₹70 per kilogram. Assume Estu weighs 50 kg and wheat costs ₹50 per kilogram. Find the value of each donation if each person donates their weight in the stated food. [4 marks]

1. For jaggery, multiply Roxie's weight by the price per kilogram: 45 × 70 rupees.
2. The value of the jaggery donation is ₹3150.
3. For wheat, multiply Estu's weight by the price per kilogram: 50 × 50 rupees.
4. The value of the wheat donation is ₹2500.

Q8. Compare the approximate distances 1.4335 × 10¹² m from Sun to Saturn, 1.439 × 10¹² m from Saturn to Uranus, and 1.496 × 10¹¹ m from Sun to Earth. Identify the smallest and greatest. [3 marks]

1. All three coefficients are at least one and less than ten, so compare the standard-form exponents first.
2. The Sun-to-Earth distance is smallest because its exponent is eleven, whereas the other two have exponent twelve.
3. For the two remaining distances, compare coefficients. Since 1.439 exceeds 1.4335, the Saturn-to-Uranus distance is greatest among the three supplied approximations.

Key takeaways

• A positive integer exponent counts repeated factors of the base; it does not mean multiplying the base by the exponent.
• Multiplying powers with the same base adds their exponents, while dividing powers with a non-zero common base subtracts them.
• Raising a power to another power multiplies the exponents because each outer group contains the same number of inner factors.
• A non-zero number raised to zero equals one; a negative exponent represents the reciprocal of the corresponding positive power.
• Scientific notation uses a coefficient at least one and less than ten multiplied by an integer power of ten.
• Linear growth adds a fixed amount at each step, whereas exponential growth repeatedly multiplies the quantity by a fixed factor.
• Count arrangements by multiplying the choices at each position, checking whether repetitions and initial zeroes are permitted.
• State assumptions when modelling quantities and preserve approximate language when the values used in a calculation are estimates.

Test yourself

Why is 4 × 4 × 4 written as 4³?

There are three repeated factors of four, so four is the base and three is the exponent.

What is (−2)⁴, and why is it positive?

It equals sixteen. Multiplying four negative factors gives a positive result because the negative signs pair up.

Simplify p⁴ × p⁶, where p represents a number.

The result is p¹⁰. Both powers have the same base, so add their exponents.

Why do (4³)² and (4²)³ give the same value?

Both contain six factors of four, so both equal 4⁶, which is 4096.

What does 10⁻³ mean?

It means the reciprocal of 10³, namely one divided by one thousand.

Which expression is standard form for 5900: 59 × 10² or 5.9 × 10³?

Use 5.9 × 10³ because its coefficient is at least one and less than ten.

A pond's lotus count doubles daily and it is fully covered on day thirty. When should it be half covered?

It should be half covered on day twenty-nine, because the next day's doubling produces full coverage.

Why should an ants-per-human calculation based on estimated populations retain “about”?

The quotient depends on approximate inputs, so performing arithmetic does not turn it into an exact population ratio.