Laws of Indices: Rules, Formulas and Examples

Laws of Indices are rules used to simplify expressions containing powers or exponents. They cover multiplication and division of powers, powers raised to powers, zero and negative exponents, and fractional indices. Applying the correct exponent rule reduces lengthy calculations and helps solve indices questions accurately.

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Overview Laws of Indices Formula & Tricks Laws of Indices Concepts Laws of Indices Video Lessons Practice Laws of Indices Questions Laws of Indices Quick Quiz Laws of Indices Revision Notes Laws of Indices FAQs

What Are the Laws of Indices?

Laws of indices are algebraic rules for simplifying expressions with exponents. An index shows how many times a number or algebraic quantity is multiplied by itself, as in a^n.

In a^n, a is the base and n is the exponent or index. For positive integers, a^n means a multiplied by itself n times. The rules allow powers with the same base to be combined or rewritten. For division rules, the base must be non-zero; fractional-index rules are generally stated for positive bases in real-number calculations.

Laws of Indices Formula & Tricks

Important Formulas

Product Rule

a^m × a^n = a^(m+n)

When powers with the same non-zero base are multiplied, add their exponents.

Quotient Rule

a^m ÷ a^n = a^(m−n), a ≠ 0

When powers with the same base are divided, subtract the denominator exponent from the numerator exponent.

Power of a Power

(a^m)^n = a^(mn)

When a power is raised to another power, multiply the exponents.

Power of a Product

(ab)^n = a^n b^n

An exponent outside brackets applies to every factor inside the brackets.

Power of a Quotient

(a/b)^n = a^n/b^n, b ≠ 0

An exponent outside a fraction applies to both the numerator and denominator.

Fractional and Negative Indices

a^(1/n) = ⁿ√a; a^(m/n) = ⁿ√(a^m); a^(−n) = 1/a^n

A fractional index represents a root, while a negative index gives the reciprocal of the corresponding positive power.

Quick Tricks

Check the Base Before Combining Powers

Add or subtract exponents only when the bases are identical. Different bases cannot be combined by the product or quotient rule.

Example: 2^3 × 2^4 = 2^7, but 2^3 × 3^4 cannot be written as one power by adding 3 and 4.

Rewrite Negative Powers as Reciprocals

Move a factor with a negative exponent across the fraction bar to make its exponent positive.

Example: x^3/y^(−2) = x^3 × y^2 = x^3y^2.

Simplify Brackets Before Expanding

For a power raised to a power, multiply the exponents directly instead of expanding the full expression.

Example: (3^2)^4 = 3^(2×4) = 3^8.

Laws of Indices Concepts

Multiplication and Division of Powers

For powers with the same base, multiplication requires addition of exponents and division requires subtraction of exponents.

Use a^m × a^n = a^(m+n) and a^m ÷ a^n = a^(m−n), where a is non-zero for division. The base remains unchanged in both rules.

Example: 5^3 × 5^2 = 5^5 = 3125, while 7^6 ÷ 7^2 = 7^4 = 2401.

Power Raised to Another Power

When an expression already containing an exponent is raised to another exponent, multiply the two exponents.

The rule is (a^m)^n = a^(mn). This also applies when the inner expression is a variable or a numerical power.

Example: (2^3)^4 = 2^(3×4) = 2^12 = 4096.

Powers of Products and Quotients

An exponent outside brackets applies to every factor in a product or to both parts of a quotient.

Use (ab)^n = a^n b^n and (a/b)^n = a^n/b^n. The quotient rule requires the denominator b to be non-zero.

Example: (2x)^3 = 2^3x^3 = 8x^3, and (3/5)^2 = 9/25.

Zero and Negative Indices

A non-zero quantity raised to the zero power equals 1, while a negative exponent represents a reciprocal.

The rules are a^0 = 1 for a ≠ 0 and a^(−n) = 1/a^n. Thus, a negative exponent does not make the value negative; it changes the number to its reciprocal.

Example: 9^0 = 1 and 2^(−3) = 1/2^3 = 1/8.