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Laws of Indices

Rules for multiplying, dividing, and raising powers, plus what a zero or negative index means.

What Is an Index?

x^{3} is read " x to the power of 3 ", or " x cubed". x^{2} is read " x squared". The index tells you how many copies of x are multiplied together.

The three main laws of indices
The three main laws of indices

These laws only work when the base (the letter or number being raised to a power) is the same on both sides.

Example 1 — multiplying

Simplify x^{2} \times x^{3} . 1 Same base (x), so add the indices: 2 + 3 = 5 2 x^{2} \times x^{3} = x^{5} Answer: x^{5}

Example 2 — dividing

Simplify x^{7} \div x^{3} . 1 Same base (x), so subtract the indices: 7 - 3 = 4 2 x^{7} \div x^{3} = x^{4} Answer: x^{4}

Example 3 — power of a power

Simplify (x^{2})^{4} . 1 Multiply the indices: 2 \times 4 = 8 2 (x^{2})^{4} = x^{8} Answer: x^{8}

Example 4 — with coefficients

Simplify 3 x^{2} \times 4 x^{3} . 1 Multiply the coefficients: 3 \times 4 = 12 2 Multiply the x -parts using the law: x^{2} \times x^{3} = x^{5} 3 Combine: 12 x^{5} Answer: 12 x^{5}

Two special cases follow directly from the division law: Any non-zero number to the power 0 equals 1 . For example, x^{0} = 1, and 7^{0} = 1 . A negative index means " 1 over" the positive power. For example, x^{- 2} means 1/ x^{2} .

Example 1 — zero index

Why does x^{0} = 1 ? 1 Using the division law: x^{3} \div x^{3} = x^{3 - 3} = x^{0} 2 But any number divided by itself is 1 : x^{3} \div x^{3} = 1 3 So x^{0} must equal 1 .

Example 2 — negative index

Simplify x^{2} \div x^{5} . 1 Subtract the indices: 2 - 5 = - 3 2 x^{2} \div x^{5} = x^{- 3} 3 This means x^{- 3} = 1 / x^{3} Answer: x^{- 3}, which is 1/ x^{3}

Practice

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Simplify x^{5} \div x^{2} .

Simplify (x^{3})^{2} and enter the resulting index (the power of x).

Your answer