Indices: Formulas, Rules and Simplification Examples

Indices represent repeated multiplication using powers or exponents. This topic covers the laws of indices, zero and negative indices, fractional powers, and methods for simplifying expressions. Learn how to combine powers with the same base, convert roots into indices, and solve numerical indices questions accurately.

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What Are Indices?

An index, or exponent, shows how many times a number or algebraic quantity is multiplied by itself. In a^n, a is the base and n is the index or exponent.

For example, 5^3 = 5 × 5 × 5 = 125. The expression a^0 equals 1 for a ≠ 0, while a^(-n) = 1/a^n. Fractional indices represent roots: a^(1/n) = the nth root of a, and a^(m/n) = (the nth root of a)^m.

Indices Formula & Tricks

Important Formulas

Product of powers
a^m × a^n = a^(m+n)

When powers have the same non-zero base, add their indices.

Quotient of powers
a^m ÷ a^n = a^(m−n), a ≠ 0

When powers have the same non-zero base, subtract the denominator index from the numerator index.

Power of a power
(a^m)^n = a^(mn)

Multiply the indices when a power is raised to another power.

Power of a product
(ab)^n = a^n b^n

An exponent applied to a product applies to each factor.

Power of a quotient
(a/b)^n = a^n/b^n, b ≠ 0

An exponent applied to a quotient applies to both numerator and denominator.

Fractional and negative indices
a^(m/n) = (ⁿ√a)^m and a^(−n) = 1/a^n

A fractional index denotes a root and a negative index denotes the reciprocal of the corresponding positive power.

Quick Tricks

Convert all terms to a common base

Rewrite numbers as powers of the same base before applying index laws.

Example: 8^2 × 4^3 = (2^3)^2 × (2^2)^3 = 2^6 × 2^6 = 2^12.
Use reciprocal form for negative powers

Move a term with a negative index across the fraction bar to make its index positive.

Example: 3^(-2) = 1/3^2 = 1/9.
Convert roots into fractional indices

Replace √a with a^(1/2) and the nth root of a with a^(1/n), then apply exponent laws.

Example: √(x^6) = (x^6)^(1/2) = x^3 for x ≥ 0.
Check the base before combining indices

Indices can be added or subtracted only when the bases are the same. Do not combine exponents across addition or subtraction.

Example: 2^3 × 3^3 = (2 × 3)^3 = 6^3, but 2^3 + 2^4 cannot be written as 2^7.

Indices Concepts

Multiplication and Division Laws

For powers with the same base, multiplication adds the indices and division subtracts the indices.

The rules are a^m × a^n = a^(m+n) and a^m ÷ a^n = a^(m−n), where a ≠ 0. These laws follow from counting repeated factors. For example, 7^5 ÷ 7^2 = 7^(5−2) = 7^3 = 343.

Example: 2^4 × 2^3 = 2^7 = 128.

Power of a Power and Distribution Laws

When a power is raised to another power, multiply the indices; when a product or quotient is raised to a power, distribute the power to each factor.

Use (a^m)^n = a^(mn), (ab)^n = a^n b^n, and (a/b)^n = a^n/b^n. For example, (3^2)^4 = 3^8, while (2 × 5)^3 = 2^3 × 5^3.

Example: (x^3/y^2)^4 = x^12/y^8, provided y ≠ 0.

Zero and Negative Indices

A non-zero quantity raised to index zero equals 1, and a negative index gives the reciprocal of the corresponding positive power.

For a ≠ 0, a^0 = 1 and a^(−n) = 1/a^n. Thus, 10^0 = 1 and 4^(−3) = 1/4^3 = 1/64. The expression 0^0 is not assigned a value in standard elementary arithmetic.

Example: x^5/x^8 = x^(5−8) = x^(−3) = 1/x^3, for x ≠ 0.

Fractional Indices and Roots

A fractional index expresses a root: a^(1/n) is the nth root of a, and a^(m/n) is the nth root of a raised to the power m.

The identities are a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)^m = (a^m)^(1/n). For real values, an even root requires a non-negative radicand. For example, 16^(3/4) = (⁴√16)^3 = 2^3 = 8.

Example: 27^(2/3) = (³√27)^2 = 3^2 = 9.

Simplifying Expressions with Different Bases

Expressions with different bases must first be rewritten using a common base before their indices can be combined.

For example, 9^2 × 3^4 becomes (3^2)^2 × 3^4 = 3^4 × 3^4 = 3^8. Exponent rules do not apply directly to addition or subtraction, so 2^3 + 2^4 remains 8 + 16 = 24.

Example: 25^(1/2) × 5^2 = 5 × 25 = 125.

Indices Video Lessons

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Indices: Power, Negative and Fractional Exponents

Understand power of a power, negative exponents, and fractional exponents in indices, including the rules used to simplify expressions accurately.

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Practice Indices Questions

Practise published questions related to this topic.

1If 5^(x+1) = 1 , what is the value of x?→ 2If 5^(x - y) = 3125 and 3^(3x + y) = 2187 , then the value of 19x - 18y is:→ 3If (3-22 + 3-19 + 3-17 + 3-16) = k(3-24), then find the value of k.→ 4Find the simplified value of the given expression. ((2³ × 8²)³ ÷ (4⁵ × 16²))/((32² ÷ 2⁴)²)→ 5If (0.04)² ÷ (0.008) × (0.2)⁶ = (0.2)^x , then find x.→ 6The simplified value of ( ∛2744/⁴√4096 ) × 4/42 × 84 is→ 7IF 2x+1 X 4y-2 = 32 and x = 3y, what is the value of y?→ 8Simplify: √75 + √48 - √27→ 9Given that 1180.65 = x, 1180.18 = y and xz = y2 , then the value of z is closest to:→ 10Simplify: √(144 × 49) + √(64 × 25) - √(100 × 36)→

Indices Quick Quiz

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Quick Revision Notes

Indices Revision Points

Use these rules while simplifying powers, roots and algebraic expressions.

  • a^m × a^n = a^(m+n) for the same base.
  • a^m ÷ a^n = a^(m−n), with a ≠ 0.
  • (a^m)^n = a^(mn).
  • (ab)^n = a^n b^n and (a/b)^n = a^n/b^n.
  • a^0 = 1 for a ≠ 0.
  • a^(−n) = 1/a^n.
  • a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)^m.
  • Rewrite different bases as powers of a common base before combining indices.

Indices FAQs

What is the product rule of indices?

For the same base, a^m × a^n = a^(m+n). For example, 5^2 × 5^3 = 5^5.

How do you divide powers with the same base?

Subtract the denominator index from the numerator index: a^m ÷ a^n = a^(m−n). Thus, 3^7 ÷ 3^4 = 3^3 = 27.

What is the value of a number with index zero?

For every non-zero number a, a^0 = 1. Therefore, 12^0 = 1.

How is a negative index converted into a positive index?

Take the reciprocal: a^(−n) = 1/a^n. For example, 2^(−4) = 1/16.

What does a fractional index mean?

The denominator represents the root and the numerator represents the power: a^(m/n) = (ⁿ√a)^m. For example, 8^(2/3) = (³√8)^2 = 4.

Can indices be added when the bases are different?

No. First express the bases in a common form, if possible. For example, 4^2 × 2^3 = (2^2)^2 × 2^3 = 2^7.

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