Exponents & Indices

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Quick recall

• Multiply same bases: add exponents. Divide same bases: subtract exponents.
• For fractional powers over the reals, check that the root is defined.

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Formula Cards

1. Multiply same bases

$$a^m a^n=a^{m+n}$$

2. Divide same bases

$$\frac{a^m}{a^n}=a^{m-n}$$

The denominator must be nonzero.

3. Power of a power

$$(a^m)^n=a^{mn}$$

4. Zero and negative powers

$$a^0=1,\qquad a^{-n}=\frac{1}{a^n}$$

Requires a ≠ 0.

5. Fractional powers

$$a^{1/n}=\sqrt[n]{a},\qquad a^{m/n}=(\sqrt[n]{a})^m$$

For real arithmetic, check that the root exists; positive bases avoid fractional-power ambiguities.

6. Power of a product

$$(ab)^n=a^nb^n$$

7. Power of a quotient

$$\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$$

Requires b ≠ 0; use values for which the powers are defined.

Worked Examples

Example 1

Question: Evaluate 2³ × 2⁴.

Step 1: Add exponents of the same base.

$$2^3\times2^4=2^{3+4}=2^7$$

Step 2: Evaluate.

$$2^7=128$$

Answer: 128.

Example 2

Question: Evaluate 5⁶ ÷ 5².

Step 1: Subtract exponents.

$$\frac{5^6}{5^2}=5^{6-2}=5^4$$

Step 2: Evaluate.

$$5^4=625$$

Answer: 625.

Example 3

Question: Evaluate 8^(2/3).

Step 1: Take the cube root first.

$$\sqrt[3]{8}=2$$

Step 2: Square the result.

$$8^{2/3}=2^2=4$$

Answer: 4.

Example 4

Question: Evaluate 27^(−1/3).

Step 1: Use the reciprocal for the negative exponent.

$$27^{-1/3}=\frac{1}{27^{1/3}}$$

Step 2: Take the cube root.

$$\frac{1}{\sqrt[3]{27}}=\frac{1}{3}$$

Answer: 1/3.

Drill

Try each question before opening its answer.

Question 1

Simplify $3^2\cdot3^5$.

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  2187

Question 2

Simplify $\dfrac{10^7}{10^4}$.

• Show answer

  1000

Question 3

Evaluate $16^{1/2}$.

• Show answer

  4

Question 4

Evaluate $4^{-2}$.

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  1/16

Question 5

Simplify $(2^3)^2$.

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  64

Question 6

Evaluate $32^{3/5}$.

• Show answer

  8

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