# StudySquare

This page covers the following topics:

1. Powers

2. Multiplying powers

3. Dividing powers

4. Power to a power

5. Power of zero

6. Fractional powers

Powers are a way of representing numbers or variables that are being multiplied by themselves. The small floating number at the top of the number or variable is called its power.

To multiply powers, the multiplication rule is used. The multiplication rule states that when terms are multiplied, the powers of the terms are added together. This rule is used on all types of powers, whether they are positive, negative or fractional.

To divide powers, the division rule is used. The division rule states that when terms are divided, the powers of the terms are subtracted.

To raise a power by a power, the power rule is used. The power rule states that to raise a power by a power, the two powers must be multiplied together.

The power of zero rule states that anything raised to the power of zero is 1.

When a term has a fractional power, it can be converted to a root. When the power is 1/b, the b-th root is taken. When the fractional power has a numerator other than 1, ie. c/b, the term is raised to the power of c and the b-th root is taken.

# 1

Simplify (3w⁶)² − (w³)⁴.

Using the power rule, (3w⁶)² − (w³)⁴ = 9w¹² − w¹² = 8w¹².

8w¹²

# 2

Show that ⁷√(w⁵) and (⁷√w)⁵ are the same.

⁷√(w⁵) = (w⁵)¹ʹ⁷ = w⁵ʹ⁷ and (⁷√w)⁵ = (w¹ʹ⁷)⁵ = w⁵ʹ⁷, so the two are the same.

5 × 1/7 = 1/7 × 5

# 3

Simplify (2s²)³ × (3s⁴p)² ÷ (2s²)².

Using the power rule and the multiplication rule, (2s²)³ × (3s⁴p)² ÷ (2s²)² = 8s⁶ × 9s⁸p² ÷ 4s⁴ = 18s¹⁰p².

18s¹⁰p²

# 4

Calculate 8¹ʹ³ × 81¹ʹ⁴.

8¹ʹ³ × 81¹ʹ⁴ = ³√8 × ⁴√81 = 2 × 3 = 6

6

# 5

Calculate 32²ʹ⁵.

32²ʹ⁵ = (⁵√32)² = 2² = 4

4

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