Working with Exponents

Working with Exponents

An exponent tells how many times a base appears as a factor. A square uses two equal factors, and a cube uses three. Exponents do not mean multiplication of the base by the exponent.

How to solve these problems

For positive integer n, aⁿ is n factors of a. For nonzero a, a⁰ = 1 and a⁻ⁿ = 1/aⁿ. Parentheses identify whether a negative sign belongs to the base.

Worked Example 1: 2³

2³

Solution: 2 × 2 × 2 = 8.

Why it works: Three is the number of factors, not a multiplier of 2.

Worked Example 2: 5²

5²

Solution: 5 × 5 = 25.

Why it works: Squaring means multiplying the base by itself.

Worked Example 3: 10⁴

10⁴

Solution: 10 × 10 × 10 × 10 = 10,000.

Why it works: A positive integer power of ten places that many zeros after 1.

Worked Example 4: (−3)²

(−3)²

Solution: (−3) × (−3) = 9.

Why it works: An even number of negative factors has a positive product.

Worked Example 5: (−2)³

(−2)³

Solution: (−2) × (−2) × (−2) = −8.

Why it works: An odd number of negative factors has a negative product.

Worked Example 6: 7⁰

7⁰

Solution: 7⁰ = 1.

Why it works: For a nonzero base, aⁿ/aⁿ = a⁰ = 1. This rule does not define 0⁰.

Worked Example 7: 2⁻³

2⁻³

Solution: 2⁻³ = 1/2³ = 1/8.

Why it works: A negative exponent indicates a reciprocal, not a negative answer.

Worked Example 8: (1/2)³

(1/2)³

Solution: 1³/2³ = 1/8.

Why it works: Raise both numerator and denominator to the power.

Worked Example 9: 0.2²

0.2²

Solution: 0.2 × 0.2 = 0.04.

Why it works: Multiplying a positive number less than one by itself makes it smaller.

Worked Example 10: Evaluate x² + 2 when x = −4

Evaluate x² + 2 when x = −4

Solution: Substitute with parentheses: (−4)² + 2 = 16 + 2 = 18.

Why it works: Parentheses preserve the sign of a substituted negative value.

Try it again

Cover the solutions and solve the ten examples on your own. Explain which rule you used, then compare your method and answer with the worked solution. Check signs, equivalent forms, and units.