Chapter 09 — Powers — Part 1: Introduction and Fractional Indices

CHAPTER 09: POWERS

PART 1: INTRODUCTION AND FRACTIONAL INDICES

In earlier mathematics, you studied powers and indices involving positive whole numbers. For example, you know that 2³ means 2 × 2 × 2.

In this chapter, we extend the meaning of indices to include zero, negative and fractional values. This allows us to express roots and powers in a shorter and more convenient form.

We shall begin by revising the meaning of indices and then develop the rules governing fractional indices.

1. What is an Index?

An index, also called an exponent or power, tells us how many times a number is used as a factor in repeated multiplication.

For example:

5³ = 5 × 5 × 5 = 125

In this expression:

• 5 is the base.
• 3 is the index or exponent.
• 125 is the value of the expression.

Generally:

aⁿ = a × a × a × ··· × a

where a is the base and n is a positive integer. There are n factors of a.

2. Revision of the Laws of Indices

Before studying fractional indices, it is important to recall the main laws of indices.

Law 1: Multiplication of powers with the same base

When multiplying powers with the same base, add their indices.

a^m × aⁿ = a^m+n

Example:

2³ × 2² = 2³⁺² = 2⁵ = 32

Law 2: Division of powers with the same base

When dividing powers with the same non-zero base, subtract the denominator's index from the numerator's index.

a^m ÷ aⁿ = a^m−n

Example:

3⁵ ÷ 3² = 3³ = 27

Law 3: Raising a power to another power

When a power is raised to another power, multiply the indices.

(a^m)ⁿ = a^mn

Example:

(2³)² = 2⁶ = 64

Law 4: Power of a product

(ab)ⁿ = aⁿbⁿ

Example:

(2 × 3)² = 2² × 3² = 36

Law 5: Power of a quotient

(a/b)ⁿ = aⁿ/bⁿ,   b ≠ 0

Example:

(2/3)² = 2²/3² = 4/9

3. Zero as an Index

Any non-zero number raised to the power zero is equal to one.

a⁰ = 1,   a ≠ 0

We can understand this using the division law of indices.

Consider:

a³ ÷ a³ = a³⁻³ = a⁰

But any non-zero number divided by itself is one:

a³ ÷ a³ = 1

Therefore:

a⁰ = 1

Example 1

Evaluate:

7⁰ + 3⁰ + 2⁰

Solution

= 1 + 1 + 1

= 3

Important: The expression 0⁰ is not assigned a value by the ordinary zero-index law. Do not apply a⁰ = 1 when a = 0.

4. Negative Indices

A negative index represents the reciprocal of the corresponding positive power.

a⁻ⁿ = 1/aⁿ,   a ≠ 0

For example:

2⁻³ = 1/2³ = 1/8

Another example:

5⁻² = 1/5² = 1/25

A negative index does not mean that the value itself is negative. It means that we take the reciprocal.

Example 2

Evaluate:

3⁻² + 2⁻³

Solution

= 1/3² + 1/2³

= 1/9 + 1/8

= 8/72 + 9/72

= 17/72

5. Meaning of Fractional Indices

A fractional index represents a root. In particular, an index of 1/n represents the nth root.

a^1/n = ⁿ√a

For example:

16^1/2 = √16 = 4

27^1/3 = ∛27 = 3

81^1/4 = ⁴√81 = 3

The denominator of the fractional index tells us which root to take.

Understanding the square root

The square root of a positive number is the non-negative number which, when squared, gives the original number.

For example:

√25 = 5 because 5² = 25

Understanding the cube root

The cube root of a number is the number which, when cubed, gives the original number.

∛64 = 4 because 4³ = 64

Remember:

• The square root is a root of order 2.
• The cube root is a root of order 3.
• The fourth root is a root of order 4.

6. Fractional Indices with Numerators Other Than One

When the index is a fraction with a numerator other than one, we first take the root indicated by the denominator and then raise the result to the power indicated by the numerator.

a^m/n = (ⁿ√a)^m = ⁿ√(a^m)

For positive real a, this gives the usual real-valued interpretation.

Example 3

Evaluate:

8^2/3

Solution

The denominator is 3, so take the cube root. The numerator is 2, so square the result.

8^2/3 = (∛8)²

= 2²

= 4

Example 4

Evaluate:

16^3/4

Solution

16^3/4 = (⁴√16)³

= 2³

= 8

Example 5

Evaluate:

125^2/3

Solution

125^2/3 = (∛125)²

= 5²

= 25

7. Negative Fractional Indices

A negative fractional index combines the rules for negative indices and fractional indices.

a^−m/n = 1/a^m/n

We take the appropriate root and power, then find the reciprocal.

Example 6

Evaluate:

27^−2/3

Solution

27^−2/3 = 1/27^2/3

= 1/(∛27)²

= 1/3²

= 1/9

Example 7

Evaluate:

16^−3/4

Solution

= 1/16^3/4

= 1/(⁴√16)³

= 1/2³

= 1/8

8. Simplifying Expressions Involving Fractional Indices

The laws of indices continue to apply when the indices are fractions, provided the expressions are defined.

Example 8

Simplify:

x^1/2 × x^3/2

Solution

Add the indices because the bases are the same.

= x^{1/2 + 3/2}

= x^4/2

= x²

Example 9

Simplify:

a^5/3 ÷ a^2/3

Solution

Subtract the indices.

= a^{5/3 − 2/3}

= a^3/3

= a

Example 10

Simplify:

(x^2/3)³

Solution

Multiply the indices.

= x^{2/3 × 3}

= x²

Example 11

Simplify:

16^1/2 × 8^1/3

Solution

= √16 × ∛8

= 4 × 2

= 8

9. Converting Between Surds and Fractional Indices

Surds and fractional indices are two ways of expressing roots. Converting between them is useful when simplifying complicated expressions.

Fractional index	Surd form
a1/2	√a
a1/3	∛a
a1/4	⁴√a
a2/3	(∛a)²
a3/4	(⁴√a)³
a−1/2	1/√a

Example 12

Express the following in fractional index form:

√x × ³√x

Solution

= x^1/2 × x^1/3

= x^{1/2 + 1/3}

= x^{3/6 + 2/6}

= x^5/6

10. Important Restrictions and Common Errors

• Division by zero is not permitted.
• A negative index means reciprocal, not a negative answer.
• The denominator of a fractional index indicates the root.
• The numerator indicates the power.
• For real-number work, an even root of a negative number is not real.
• When using fractional powers of variables, be aware of the restrictions needed for real-valued expressions.
• Do not add indices when the bases are different.

11. Exercises

Attempt the following questions before opening the worked solutions. Show each step clearly.

Question 1

Evaluate:

(a) 5⁰

(b) 2⁻³

(c) 3⁻²

Show worked solutions

(a) 5⁰ = 1

(b) 2⁻³ = 1/2³ = 1/8

(c) 3⁻² = 1/3² = 1/9

Question 2

Evaluate:

(a) 25^1/2

(b) 64^1/3

(c) 81^1/4

Show worked solutions

(a) √25 = 5

(b) ∛64 = 4

(c) ⁴√81 = 3

Question 3

Evaluate:

(a) 8^2/3

(b) 32^2/5

(c) 81^3/4

Show worked solutions

(a) (∛8)² = 2² = 4

(b) (⁵√32)² = 2² = 4

(c) (⁴√81)³ = 3³ = 27

Question 4

Evaluate:

(a) 16^−1/2

(b) 8^−2/3

(c) 81^−3/4

Show worked solutions

(a) 1/√16 = 1/4

(b) 1/(∛8)² = 1/4 = 1/4

(c) 1/(⁴√81)³ = 1/27 = 1/27

Question 5

Simplify:

(a) x^2/3 × x^1/3

(b) y^7/4 ÷ y^3/4

(c) (a^2/5)³

Show worked solutions

(a) x^2/3+1/3 = x

(b) y^7/4−3/4 = y = y

(c) a^2/5×3 = a^6/5

Question 6

Express each expression as a fractional index:

(a) √x

(b) ∛(x²)

(c) 1/⁴√x

Show worked solutions

(a) x^1/2

(b) x^2/3

(c) x^−1/4

Question 7

Express each expression in surd form:

(a) x^3/2

(b) a^2/3

(c) y^−1/3

Show worked solutions

(a) (√x)³

(b) (∛a)²

(c) 1/∛y

Question 8

Simplify:

(a) 27^1/3 × 16^1/2

(b) 32^3/5 ÷ 9^1/2

Show worked solutions

(a) 3 × 4 = 12

(b) 2³ ÷ 3 = 8/3

Question 9

Simplify fully:

(x^3/4 × x^1/2) ÷ x^1/4

Show worked solution

= x^{3/4 + 1/2 − 1/4}

= x^{3/4 + 2/4 − 1/4}

= x^4/4

= x

Question 10

Evaluate:

16^3/4 + 27^2/3 − 32^1/5

Show worked solution

= 8 + 9 − 2

= 15

12. Lesson Summary

Zero index

a⁰ = 1, a ≠ 0

Negative index

a⁻ⁿ = 1/aⁿ

Fractional index

a^1/n = ⁿ√a

General fractional index

a^m/n = (ⁿ√a)^m

Negative fractional index

a^−m/n = 1/a^m/n

By completing this lesson, you should be able to:

• Explain the meaning of zero, negative and fractional indices.
• Evaluate numerical expressions involving fractional indices.
• Convert between surds and fractional index notation.
• Simplify algebraic expressions involving fractional indices.
• Apply the laws of indices correctly.

End of Part 1

You have completed the introduction to powers and fractional indices.

Next: Part 2 — Further Problems and Simplification of Indices.

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