Indices/ Exponents | ICSE Class 9 Maths Notes

This note covers indices and exponential notation, positive and zero indices, negative and fractional indices, the laws of exponents, powers of products and quotients, simplification of expressions, and equations involving powers with a common base.

What do a base, an index and a power mean?

An index, also called an exponent, is the raised number that specifies a power. Its plural is indices. The base is the number being raised to that power. For a positive integer exponent, the base is multiplied by itself the stated number of times.

A positive integer is a whole number greater than zero. In 2⁵, the base is 2 and the exponent is 5. The expression means 2 × 2 × 2 × 2 × 2. Each multiplied number is called a factor.

Definition: Let a represent a number and n a positive integer. The power aⁿ means the product of n factors, each equal to a. A product is the result of multiplication.

How should exponential notation be read?

Read aⁿ as “a raised to the power n”. The exponent counts factors, not additions. Thus, writing a number as a power records a repeated multiplication compactly. It does not instruct you to multiply the base by the exponent.

Worked example 1. Expand 2⁵ as repeated multiplication and evaluate it.

Answer: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. There are five factors of 2. The raised 5 tells you how many copies of the base appear in the product.

Brackets show the complete base when it contains a sign or a fraction. For instance, (−4)⁵ has the negative integer −4 as its base. The minus sign belongs to every repeated factor because it is inside the brackets.

The repeated-multiplication definition directly explains positive integer indices. Zero, negative and fractional indices need further definitions. These definitions are chosen so that the laws of exponents continue to work under their stated conditions. Do not interpret a negative index as a negative count of factors.

How are powers with the same base multiplied?

Property: multiplication of powers with a common base

Let a be a non-zero number, meaning a number other than zero, and let m and n be integers. An integer is a positive or negative whole number, or zero. The multiplication law is aᵐ × aⁿ = a⁽ᵐ⁺ⁿ⁾. Keep the common base and add the exponents.

For positive integer indices, the reason comes from counting repeated factors. The first power supplies m copies of a, and the second supplies n more. Their product therefore contains m + n copies of the same base.

The law also works for negative integer indices, using the reciprocal meaning explained below. A reciprocal is a number which gives 1 when multiplied by the original non-zero number. Signed exponents must be added with their signs intact.

Worked example 2. Simplify 5⁻² × 5⁴.

Answer: The bases are both 5, so add the exponents: 5⁻² × 5⁴ = 5⁽⁻²⁺⁴⁾ = 5². The negative exponent contributes −2 to the sum; it does not make the final power negative.

What changes when the common base is negative?

The same operation applies to a negative base with integer exponents. Keep brackets around the base throughout the calculation. Otherwise, the written expression may no longer show whether the minus sign belongs to the base or stands outside the power.

Worked example 3. Simplify (−4)⁵ × (−4)⁻¹⁰.

Answer: (−4)⁵ × (−4)⁻¹⁰ = (−4)⁽⁵⁻¹⁰⁾ = (−4)⁻⁵ = 1/(−4)⁵. The base stays −4, while the exponent becomes −5. The final expression uses a positive index in the denominator.

A denominator is the number below the fraction bar; the number above it is the numerator. These examples combine powers through multiplication. An addition sign between powers gives a different operation, so the same-base multiplication law cannot simply be used across a sum.

How are powers with the same base divided?

Property: division of powers with a common base

For non-zero a and integer exponents m and n, aᵐ ÷ aⁿ = a⁽ᵐ⁻ⁿ⁾. Division produces a quotient. Keep the base and subtract the exponent of the divisor, the power by which you are dividing, from the exponent of the dividend.

The dividend is the quantity being divided. This order matters: the exponent belonging to the dividend comes first. If the divisor has a negative exponent, put that exponent in brackets before subtracting it. Subtracting a negative number becomes addition.

Worked example 4. Simplify 2⁵ ÷ 2⁻⁶.

Answer: 2⁵ ÷ 2⁻⁶ = 2⁽⁵⁻⁽⁻⁶⁾⁾ = 2¹¹. The exponent calculation is 5 − (−6) = 11. It is the complete exponent −6 that is subtracted, so writing 5 − 6 would change the calculation.

Why must the base be non-zero?

Division by zero is undefined. The non-zero condition makes the denominator a valid divisor. With positive integer exponents, cancellation of repeated factors explains the law: equal non-zero factors above and below the fraction bar contribute a factor of 1.

If the exponent left after subtraction is negative, keep that result or express it as a reciprocal with a positive exponent. A negative result from subtracting indices is not a reason to reverse the subtraction or change the base.

This comparison separates three operations that are easy to confuse. Identify the operation in the expression before deciding what to do to its indices. The written brackets and operation signs determine the applicable law.

What happens when a power is raised to another power?

Property: a power of a power

For non-zero a and integer exponents m and n, (aᵐ)ⁿ = aᵐⁿ. Here mn means m multiplied by n. The brackets show that the outer exponent acts on the complete inner power. Multiply the two indices, keeping their signs.

For positive integer exponents, each of the n copies of aᵐ contributes m factors of a. Altogether there are m × n factors. This explains why multiplication of indices belongs to a power of a power, rather than to a product of separate powers.

How can this law help change the base?

Sometimes a base can itself be written as a power of a simpler number. Substitute that expression first, then multiply the indices. This is called writing powers with a common base, meaning the same base in each expression being compared or combined.

Worked example 5. Express 4⁻³ as a power with base 2.

Answer: Since 4 = 2², write 4⁻³ = (2²)⁻³. Multiply the indices to obtain 2⁽²×⁽⁻³⁾⁾ = 2⁻⁶. The required base is 2, and the final index is −6.

In this calculation, replacing 4 by 2² preserves the value of the base. The outer index −3 still applies to the whole replacement. Leaving out the brackets would hide the relationship between the inner power and the outer exponent.

Note: In a product of powers with the same base, add indices. In a power of a power, multiply indices. These are different structures, even when they contain the same base and the same exponent symbols.

Keep the intermediate bracketed line when changing a base. It makes the reason for multiplying indices visible and allows the sign calculation to be checked independently of the base conversion.

Why does a non-zero number raised to zero equal one?

The zero-index rule is a⁰ = 1 for a non-zero base a. A zero index does not mean multiplying the base by zero. It belongs to the extension of exponential notation beyond its original repeated-multiplication meaning.

How does division explain the zero-index rule?

1. Take a non-zero base a and a positive integer n, so that the power aⁿ is non-zero.
2. Divide that power by itself. The quotient aⁿ ÷ aⁿ equals 1.
3. Apply the division law to the same quotient, giving a⁽ⁿ⁻ⁿ⁾.
4. Since n − n = 0, the power is a⁰. Thus a⁰ = 1 for the stated non-zero base.

The condition on the base is part of the rule. It cannot be discarded after the calculation. This argument relies on a number being divided by itself, which requires that the number being divided by is not zero.

The pattern of decreasing powers gives the same result. In the sequence 3³ = 27, 3² = 9 and 3¹ = 3, each decrease of one in the exponent divides the value by 3. The next value is therefore 3⁰ = 1.

Can a whole expression have a zero index?

Yes. If brackets surround an expression, that entire expression is the base. First establish that it is non-zero. It can then be raised to zero without calculating every part of its exact value.

A term is a part of an expression separated from other parts by addition or subtraction at that level of the expression.

Worked example 6. Evaluate (3⁻¹ + 4⁻¹ + 5⁻¹)⁰.

Answer: The bracket contains 1/3 + 1/4 + 1/5. Each term is positive, so their sum is positive and therefore non-zero. The whole non-zero bracket is raised to zero, giving 1.

Here the zero-index rule acts on the entire sum, not on each term separately. The sum is treated as one complete base.

What does a negative index mean?

For a non-zero base a and a positive integer m, a⁻ᵐ = 1/aᵐ. A negative index means taking the reciprocal of the corresponding positive power. It does not, by itself, say that the value of the expression is negative.

The multiplicative inverse is another name for the reciprocal. The powers aᵐ and a⁻ᵐ multiply to a⁰, which is 1. This connects the negative-index definition directly to the multiplication law and the zero-index rule.

Worked example 7. Evaluate 2⁻³ and 1/(3⁻²).

Answer: 2⁻³ = 1/2³ = 1/8. Also, 1/(3⁻²) = 3² = 9. Taking the reciprocal of a reciprocal restores the corresponding positive power in the second expression.

How does a negative power affect a fraction?

Let b also represent a non-zero number. For a fraction a/b with both a and b non-zero, (a/b)⁻ᵐ = (b/a)ᵐ. Invert the entire fraction and make the exponent positive. The brackets show that the power belongs to the whole fraction.

Worked example 8. Find the value of (2/3)⁻².

Answer: Invert the fraction to write (2/3)⁻² = (3/2)². Then square, meaning raise to the power 2, the numerator and denominator: (3/2)² = 3²/2² = 9/4. The value is positive.

For an integer exponent and a negative base, retain the sign inside the base brackets. An even integer is divisible by 2; an odd integer is not. An even power of a negative base is positive, while an odd power is negative.

Taking a reciprocal preserves that sign. Thus, first distinguish the minus sign in the base from the minus sign in the index. One affects the repeated factors; the other instructs you to take a reciprocal.

How do equal exponents help with different bases?

Property: a common power of a product or quotient

For non-zero bases a and b and an integer exponent m, aᵐ × bᵐ = (ab)ᵐ. Here ab means a multiplied by b. Equal exponents allow the bases to be multiplied while the shared exponent is retained.

Similarly, aᵐ ÷ bᵐ = (a/b)ᵐ when the expressions are defined and b is non-zero. These laws can be used in either direction: combine bases under one power, or apply a power to each factor or to the numerator and denominator.

This is different from the same-base law. With a common base, add the indices of multiplied powers. With a common exponent, multiply the bases and keep the exponent. Check which part of the written powers is actually the same.

Worked example 9. Simplify (−4)⁻³ × 5⁻³ × (−5)⁻³.

Answer: All three indices are −3, so combine the bases: [(−4) × 5 × (−5)]⁻³ = 100⁻³ = 1/100³. The two negative factors in the combined base give a positive product.

Can a fraction be rewritten to reveal a common exponent?

An expression may not initially display matching indices. A numerical fraction can sometimes be written as a power, revealing a common exponent. This converts the expression into a form where the common-exponent law can be applied directly.

Worked example 10. Simplify (1/8) × 3⁻³ in exponential form.

Answer: Since 8 = 2³, write 1/8 = 2⁻³. Therefore, 2⁻³ × 3⁻³ = (2 × 3)⁻³ = 6⁻³ = 1/6³. Both factors now have the same exponent.

These rules distribute powers across multiplication and division. They do not distribute powers across addition or subtraction. If a bracket contains a sum, identify and evaluate the terms or use an appropriate algebraic identity, an equality valid for all permitted values of its variables. A variable is a symbol representing a number. Do not treat the sum as a product.

How are fractional indices connected with roots?

A rational number can be written as p/q, where p and q are integers and q is non-zero. In the fractional indices used here, take q positive. A real number is a rational or irrational number; an irrational number cannot be expressed as such an integer fraction.

For a positive base a, the fractional power a⁽¹⁄ᑫ⁾ means the positive qth root of a. This root is the positive number which, raised to the positive integer power q, gives a. The symbol √ denotes the non-negative square root; ∛ denotes the cube root.

What do the numerator and denominator of an index do?

The denominator q selects the root, and the numerator p gives the power: a⁽ᵖ⁄ᑫ⁾ = (a⁽¹⁄ᑫ⁾)ᵖ. With a positive base, you can take the root first and then raise it to the numerator's power. A negative numerator also requires a reciprocal.

The positive-base condition lets the rational-index laws be applied consistently. It includes square roots, cube roots and higher roots without having to treat negative bases separately. Do not transfer an integer-index calculation with a negative base to an arbitrary fractional index without checking its meaning.

Worked example 11. Evaluate (1/27)⁽⁻²⁄³⁾.

Answer: The negative index gives (1/27)⁽⁻²⁄³⁾ = 27⁽²⁄³⁾. Since 27 = 3³, its positive cube root is 3. Squaring this root gives 3² = 9.

How can decimal exponents be handled?

A terminating decimal has finitely many digits after its decimal point and represents a rational number. Such an exponent can be added as a decimal and then expressed as a fraction to identify the required root.

Worked example 12. Evaluate 256⁰·¹⁶ × 256⁰·⁰⁹, where the raised decimal exponents are 0.16 and 0.09.

Answer: Add the exponents because the bases are equal: 256⁽⁰·¹⁶⁺⁰·⁰⁹⁾ = 256⁰·²⁵ = 256⁽¹⁄⁴⁾ = 4. The final step takes the positive fourth root of 256.

How are several fractional powers simplified together?

The familiar exponent laws extend to rational indices when the bases are positive. For a positive base a and rational numbers p and q, multiplication adds p and q, division subtracts q from p, and a power of a power multiplies p by q.

For these laws, p and q stand for complete exponents. They need not be the numerator and denominator of a single fraction. Read each formula with the meaning assigned to its letters. A symbol can represent different quantities in separately defined expressions.

How should a bracket containing addition be approached?

Evaluate the fractional powers within the bracket first. A sum of roots does not become a root of the sum by an exponent law. After finding the inner sum, deal with its power, any outside factor, and the outermost power in that order.

Worked example 13. Simplify [5(8⁽¹⁄³⁾ + 27⁽¹⁄³⁾)³]⁽¹⁄⁴⁾.

Answer: The cube roots are 2 and 3. Hence the expression becomes [5(2 + 3)³]⁽¹⁄⁴⁾ = [5 × 5³]⁽¹⁄⁴⁾ = [5⁴]⁽¹⁄⁴⁾ = 5. Addition is completed before the product is combined into a power.

The outermost power applies to everything inside the square brackets. In this example the factor 5 is part of that base, so it must be included before the fourth root is taken. The exponent laws work on the complete expressions enclosed by brackets.

How should fractional powers joined by subtraction be handled?

Subtraction separates terms just as addition does. Simplify each term independently unless the expression is first rewritten as a product. Keep the operation between terms throughout the working, rather than merging all visible exponents into one calculation.

Worked example 14. Simplify 64⁽⁻¹⁄³⁾ × 64⁽¹⁄³⁾ − 64⁽²⁄³⁾.

Answer: Combine the product first: 64⁽⁻¹⁄³⁾ × 64⁽¹⁄³⁾ = 64⁰ = 1. The positive cube root of 64 is 4, so 64⁽²⁄³⁾ = 4² = 16. The complete expression is 1 − 16 = −15.

This example shows why recognising the structure of an expression can save arithmetic. The product becomes 1 immediately because its indices add to zero. The remaining term is evaluated separately before subtraction.

How can a complete simplification or index equation be organised?

A reliable simplification starts with the brackets and operation signs. Identify each base and its complete exponent, including any minus sign. Then choose one valid law at a time, preserving the other parts of the expression until they are needed.

What sequence keeps a mixed calculation clear?

1. Identify brackets and decide which operation must be completed inside them.
2. Rewrite suitable bases as powers of a common base, without changing the expression's value.
3. Apply the division, multiplication or power-of-a-power law to the relevant parts.
4. Combine the remaining powers and express negative indices as reciprocals if positive indices are requested.

Worked example 15. Simplify (2⁵ ÷ 2⁸)⁵ × 2⁻⁵ and give an answer with a positive index.

Answer: Inside the brackets, 2⁵ ÷ 2⁸ = 2⁻³. Raising this to 5 gives (2⁻³)⁵ = 2⁻¹⁵. Multiplication by 2⁻⁵ gives 2⁻²⁰, so the requested form is 1/2²⁰.

When can exponents on opposite sides be equated?

An equation states that two expressions have equal values. For a positive common base other than 1, equality of two powers implies equality of their rational exponents. This allows an equation involving an unknown index to be reduced to an ordinary equation.

Worked example 16. Find the unknown exponent m if 5ᵐ ÷ 5⁻³ = 5⁵.

Answer: The division law gives 5⁽ᵐ⁻⁽⁻³⁾⁾ = 5⁵, or 5⁽ᵐ⁺³⁾ = 5⁵. The common base is 5, so m + 3 = 5 and m = 2. Substitution gives 5⁽²⁺³⁾ = 5⁵.

The condition on the common base matters. Powers of 1 remain 1 for different exponents. Integer powers of −1 repeat values too. Therefore, equal written bases alone are not enough to justify equating indices in every possible equation.

Finish by checking the requested form. An exact reciprocal, a single power with a specified base, and an evaluated number can represent the same value, but the instruction determines which form should appear at the end.

Glossary

• Base — The number or complete bracketed expression that is raised to a stated power.
• Index — The raised number specifying a power; also called an exponent, with plural indices.
• Positive integer power — A product containing the stated positive whole number of identical factors of its base.
• Factor — A number or expression multiplied by another to form a product.
• Quotient — The result of dividing one quantity by another non-zero quantity.
• Reciprocal — The multiplicative inverse of a non-zero number, giving one when multiplied by that number.
• Zero index — An exponent of zero, giving the value one when its base is non-zero.
• Negative index — An index indicating the reciprocal of the corresponding positive-index power of a non-zero base.
• Rational index — An exponent expressible as a fraction of integers with a non-zero denominator.
• Positive qth root — The positive number whose qth power equals the given positive number, with q a positive integer.
• Common base — The same base appearing in powers that are being combined or compared.
• Power of a power — An expression formed by raising an entire existing power to another exponent.

Common errors and misconceptions

• Misconception: The exponent in 2⁵ means multiplying 2 by 5. Correct: It means multiplying five factors of 2, giving 32.
• Misconception: Multiply the indices in 5⁻² × 5⁴. Correct: The powers are multiplied, so add their indices: −2 + 4 = 2.
• Misconception: The index in 2⁵ ÷ 2⁻⁶ is 5 − 6. Correct: Subtract the complete divisor index: 5 − (−6) = 11.
• Misconception: A zero exponent makes the value zero. Correct: A non-zero base raised to zero equals 1. The condition on the base matters.
• Misconception: A negative index makes a positive base negative. Correct: It takes a reciprocal. For example, 2⁻³ = 1/8, which is positive.
• Misconception: A fractional index means dividing the base by its denominator. Correct: That denominator selects a root; the numerator supplies the power.
• Misconception: A power can be distributed across any sum in brackets. Correct: The product and quotient laws do not apply across addition. Preserve the sum and evaluate it appropriately.
• Misconception: Equal bases justify equating indices without conditions. Correct: For positive real bases this step requires a base other than 1. Powers of 1 can agree with different indices.

Exam-style questions with model answers

Q1. Evaluate 2⁻³, showing how the negative index is handled. [2 marks]

1. Use the negative-index definition to write 2⁻³ = 1/2³, keeping the positive base 2 unchanged.
2. Since 2³ = 8, the value is 1/8. The negative index gives a reciprocal, not a negative answer.

Q2. Simplify 2⁵ ÷ 2⁻⁶ as one power of 2, explaining the index calculation. [3 marks]

1. Both powers have the same non-zero base 2. Use the division law, which subtracts the divisor's exponent from the dividend's exponent.
2. Retain the sign of the divisor's exponent by writing 5 − (−6). Subtracting this negative number gives 5 + 6 = 11.
3. Keep the common base unchanged. The simplified expression is therefore 2¹¹, already written as the required single power of 2.

Q3. Evaluate (2/3)⁻². Show the reciprocal step and the evaluation of the resulting fraction. [3 marks]

1. The entire fraction 2/3 is the base because it is enclosed in brackets. A negative power requires its reciprocal, giving (3/2)².
2. Apply the square to both parts of the fraction: (3/2)² = 3²/2². This uses the power-of-a-quotient law with a non-zero denominator.
3. Evaluate the numerator as 9 and the denominator as 4. The exact value of the original expression is 9/4.

Q4. Evaluate (1/27)⁽⁻²⁄³⁾, explaining the roles of the negative sign and the fractional index. [4 marks]

1. The base 1/27 is positive, so the rational-index laws apply. The negative exponent first gives the reciprocal base: (1/27)⁽⁻²⁄³⁾ = 27⁽²⁄³⁾.
2. The denominator 3 in the exponent requires a cube root. Write the base as 27 = 3³.
3. The positive cube root of 27 is 3. The numerator 2 in the exponent then requires the square of this root.
4. Consequently, 27⁽²⁄³⁾ = 3² = 9. This is the value of the original negative fractional power.

Q5. Simplify (2⁵ ÷ 2⁸)⁵ × 2⁻⁵. Give the result as a single power and then with a positive index. [5 marks]

1. Begin with the division inside the brackets. Both powers have base 2, so the division law gives 2⁵ ÷ 2⁸ = 2⁽⁵⁻⁸⁾.
2. Subtract the indices to obtain 2⁻³ inside the brackets. The complete expression is now (2⁻³)⁵ × 2⁻⁵, with the outside factor still present.
3. Apply the power-of-a-power law to the bracketed power. Multiply −3 by 5, giving (2⁻³)⁵ = 2⁻¹⁵.
4. Multiply the remaining powers with common base 2 by adding their indices: 2⁻¹⁵ × 2⁻⁵ = 2⁻²⁰. This is the single-power form.
5. Use the reciprocal definition of a negative index. The equivalent expression with a positive index is 1/2²⁰, which completes the requested forms.

Q6. Find the unknown exponent m in 5ᵐ ÷ 5⁻³ = 5⁵, and check your result by substitution. [4 marks]

1. The left side contains powers with the same non-zero base. Apply the division law to obtain 5⁽ᵐ⁻⁽⁻³⁾⁾ = 5⁵.
2. Simplify the exponent to m + 3. Since the common base 5 is positive and different from 1, equate the exponents: m + 3 = 5.
3. Subtract 3 from both sides of this ordinary equation. The unknown exponent is therefore m = 2.
4. Substitute 2 into the original left side: 5² ÷ 5⁻³ = 5⁽²⁺³⁾ = 5⁵. This equals the given right side.

Q7. Simplify [5(8⁽¹⁄³⁾ + 27⁽¹⁄³⁾)³]⁽¹⁄⁴⁾, showing the order in which the roots and powers are handled. [5 marks]

1. Evaluate the two cube roots within the inner brackets. The positive cube root of 8 is 2, and that of 27 is 3.
2. Add these root values inside the brackets: 2 + 3 = 5. The complete expression becomes [5 × 5³]⁽¹⁄⁴⁾.
3. The factor 5 is the first power of 5. Combine it with 5³ by adding the indices, giving 5 × 5³ = 5⁴.
4. The outer exponent now acts on the whole bracketed power, so the expression is (5⁴)⁽¹⁄⁴⁾. Multiply the indices to obtain 5⁽⁴×¹⁄⁴⁾.
5. The resulting exponent is 1, so the answer is 5. The positive base makes the fourth-root step valid and gives the positive result.

Q8. Evaluate 256⁰·¹⁶ × 256⁰·⁰⁹, where the exponents are the decimals 0.16 and 0.09. [3 marks]

1. The two factors have common positive base 256. Add their exponents to get 0.16 + 0.09 = 0.25.
2. Express this decimal index as the fraction 1/4. The product is therefore 256⁽¹⁄⁴⁾, which means the positive fourth root of 256.
3. Since 4⁴ = 256, that fourth root is 4. Hence the given product has the exact value 4.

Key takeaways

• A positive integer index counts repeated factors of the base; brackets identify the complete base when signs or fractions are involved.
• For multiplied powers with a common base, add indices; for division, subtract the divisor's index from the dividend's index.
• A power raised to another power requires multiplication of the indices, including their signs.
• A non-zero base raised to zero equals one; the non-zero condition is essential to the division argument.
• A negative index means a reciprocal. Invert an entire fractional base before using the corresponding positive exponent.
• With positive bases, a fractional index selects a root through its denominator and a power through its numerator.
• Equal exponents can combine different bases under a common power, but product laws do not distribute across sums.
• Preserve brackets, simplify one operation at a time, and check that the final answer has the form requested.

Test yourself

What are the base and exponent in 2⁵?

The base is 2 and the exponent is 5. The power represents five factors of 2 multiplied together.

Which operation on indices simplifies 5⁻² × 5⁴?

Add the indices because the powers are multiplied and their bases match. The resulting power is 5².

Why is the index in 2⁵ ÷ 2⁻⁶ equal to 11?

The division law gives 5 − (−6), which equals 11. The negative sign belongs to the exponent being subtracted.

How is 4⁻³ expressed with base 2?

Write 4 = 2², then apply the power-of-a-power law: (2²)⁻³ = 2⁻⁶.

What condition must be checked before using a⁰ = 1?

The base a must be non-zero. The rule follows from dividing a non-zero power by itself.

What does the denominator 3 mean in 27⁽²⁄³⁾?

It selects the cube root of the positive base. The numerator 2 then requires squaring that root.

Why cannot all exponents in 64⁽⁻¹⁄³⁾ × 64⁽¹⁄³⁾ − 64⁽²⁄³⁾ be added?

The subtraction separates two terms. Combine only the product first to get 1, then subtract 16 to obtain −15.

Why is a common base of 1 unsuitable for equating unknown indices?

Different exponents can give the same power of 1. Equality of those powers therefore does not establish equality of their indices.