Quantitative Aptitude · Ratio and Proportion, Indices and Logarithms

Laws of Indices for CA Foundation

Updated 1 October 2026 · Fact-checked

Indices (exponents) show how many times a base is multiplied by itself. Laws of indices let you simplify powers with the same base: add powers when multiplying, subtract when dividing, multiply when raising a power to a power. To solve exponential equations, write both sides with the same base and equate the powers.

Understand Laws of Indices

An index (also called exponent or power) tells you how many times to multiply a number by itself. In 2⁵, the base is 2 and the index is 5. So 2⁵ = 2 × 2 × 2 × 2 × 2 = 32.

The laws come straight from this meaning. Take a² × a³. That is (a × a) × (a × a × a) = a⁵. You just counted the total number of a's. This is why powers are added when you multiply terms with the same base.

The same idea explains zero and negative powers. Divide a³ by a³. The answer is 1. By the subtraction law it is a³⁻³ = a⁰. So a⁰ = 1 for any a ≠ 0. Now divide a² by a³. That is 1/a, and the law gives a⁻¹. So a⁻ⁿ = 1/aⁿ. A negative power means reciprocal. It does not make the number negative.

A fractional power means a root. a^(1/2) is the square root of a. a^(1/n) is the nth root. a^(m/n) means take the nth root and raise to power m. The same laws work for these powers.

An exponential equation has the unknown in the power, like 2^x = 32. The key idea: if the bases are the same (and the base is positive and not 1), the powers must be equal. So you rewrite both sides to a common base and equate the indices.

Key formulas to remember

Product law: aᵐ × aⁿ = aᵐ⁺ⁿ; Same base only. Add the powers.
Quotient law: aᵐ ÷ aⁿ = aᵐ⁻ⁿ; Same base, a ≠ 0. Subtract the powers.
Power of a power: (aᵐ)ⁿ = aᵐⁿ; Multiply the powers.
Power of a product: (ab)ⁿ = aⁿ × bⁿ; Applies to each factor inside the bracket.
Power of a quotient: (a/b)ⁿ = aⁿ ÷ bⁿ; b ≠ 0.
Zero index: a⁰ = 1; For a ≠ 0. 0⁰ is not defined at this level.
Negative index: a⁻ⁿ = 1 ÷ aⁿ; a ≠ 0. Also (a/b)⁻ⁿ = (b/a)ⁿ.
Fractional index: a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ); For a > 0 in general. Take the root first to keep numbers small.
Equal bases rule: If aˣ = aʸ, then x = y; Valid when a > 0 and a ≠ 1.
Equal powers rule: If aˣ = bˣ with x ≠ 0, then a = b; For positive a and b.

How to solve Laws of Indices questions

Use this method for any question on indices, whether it asks you to simplify an expression or solve for x.

1Write every number in prime-factor form, such as 8 = 2³, 27 = 3³, 0.25 = 2⁻².
2Convert roots to fractional powers and negative powers to reciprocals, or the other way round, so everything is in the same style.
3Group terms with the same base. Apply the product, quotient and power-of-power laws one at a time.
4Simplify the total index of each base by adding or subtracting the fractions carefully.
5For an equation, make the bases on both sides identical. If the bases differ but the powers match, equate the bases instead.
6Equate the indices and solve the resulting linear (or quadratic) equation.
7Check the answer by substituting into the original expression, or against the options.

Quickest way: Common-base and option-substitution method

When to use it: Use it for MCQs where the options are small numbers. It saves time and guards against sign errors.

Spot the common base at once: 2, 3, 5 and 7 cover most questions.
Convert and combine indices in one line. Do not expand large powers.
In an equation, equate indices straight away and solve.
If the algebra looks long, substitute the options. Start with the easiest value, such as 0, 1 or 2.
Eliminate options that give the wrong sign, or give a fraction when the answer must be a whole number.
Skip the question if the bases cannot be made common and the options do not help. A wrong answer costs 0.25 marks.

Common mistakes in Laws of Indices

Adding the powers when the bases are different, such as 2³ × 3² = 6⁵.
Students remember the product law but forget it needs the same base.
Fix: Check the bases first. Add powers only if the bases match. Otherwise, rewrite the bases or calculate separately.
Treating a⁻ⁿ as a negative number, so 2⁻³ = −8.
The minus sign is confused with the sign of the number.
Fix: A negative index means reciprocal. 2⁻³ = 1/2³ = 1/8.
Writing (aᵐ)ⁿ = aᵐ⁺ⁿ or aᵐ × aⁿ = aᵐⁿ.
The two laws look alike and get mixed up.
Fix: Remember: multiplying terms means adding powers. A power raised to a power means multiplying powers.
Saying a⁰ = 0.
Students think zero power means nothing is left.
Fix: Use a³ ÷ a³ = a⁰ = 1. Any non-zero number to the power 0 is 1.
Writing (a + b)ⁿ = aⁿ + bⁿ.
Students copy the power-of-a-product law to a sum.
Fix: The law works only for products and quotients. For sums, expand or substitute numbers.
Equating powers when the bases are not the same, or when the base is 1.
Students rush to the equating step.
Fix: Make the bases identical first. Remember the rule needs a > 0 and a ≠ 1.

Worked examples

Example 1

The value of (2⁵ × 4³) ÷ 8² is: (a) 4 (b) 8 (c) 16 (d) 32

Show the solution

Write all terms with base 2: 4 = 2², 8 = 2³.
4³ = (2²)³ = 2⁶ and 8² = (2³)² = 2⁶.
Numerator = 2⁵ × 2⁶ = 2¹¹.
Divide: 2¹¹ ÷ 2⁶ = 2⁵ = 32.

Answer: (d) 32

Example 2

If 3^(x+1) = 81, then x is: (a) 2 (b) 3 (c) 4 (d) 5

Show the solution

Write 81 as a power of 3: 81 = 3⁴.
So 3^(x+1) = 3⁴.
Bases are equal, so equate the powers: x + 1 = 4.
x = 3.
Check: 3^(3+1) = 3⁴ = 81.

Answer: (b) 3

Example 3

The value of 16^(3/4) × 27^(−2/3) is: (a) 8/9 (b) 9/8 (c) 2/3 (d) 3/2

Show the solution

16 = 2⁴, so 16^(3/4) = 2^(4 × 3/4) = 2³ = 8.
27 = 3³, so 27^(−2/3) = 3^(3 × −2/3) = 3⁻² = 1/9.
Multiply: 8 × 1/9 = 8/9.

Answer: (a) 8/9

Exam tips

Memorise the squares, cubes and powers of 2, 3 and 5 up to about 2¹⁰, 3⁵ and 5⁴. They let you find common bases quickly.
Questions often mix roots and fractional powers. Convert every root to a fractional index before you simplify.
Watch the sign of the index. Options often include the answer and its reciprocal as traps.
When two options look like reciprocals, recheck whether a negative index was applied correctly.
Do not spend more than a minute on a long equation. Try substituting options, or skip it.

Practice questions from Ratio and Proportion, Indices and Logarithms

Laws of Indices: frequently asked questions

What are the main laws of indices for CA Foundation?

The core laws are aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, (ab)ⁿ = aⁿbⁿ, a⁰ = 1 and a⁻ⁿ = 1/aⁿ. Fractional indices represent roots. Learn each with its condition, such as a ≠ 0.

How do I solve exponential equations using indices?

Rewrite both sides with the same base, using prime factors. Then equate the powers and solve the equation. If the bases differ and the powers are equal, equate the bases instead.

Why is any number to the power zero equal to 1?

Because aⁿ ÷ aⁿ = 1 and the quotient law gives aⁿ⁻ⁿ = a⁰. So a⁰ must be 1. This holds for a ≠ 0.

Is the same law used for surds and fractional powers?

Yes. A surd such as √a is simply a^(1/2), and ⁿ√a is a^(1/n). Once you convert surds to fractional indices, all the laws of indices apply directly.