Negative and Fractional Indices

📚 Introduction

So far, you might have worked with positive whole number indices (powers), like $x^2$ or $x^3$. But what happens when we have negative indices like $x^{-2}$, or fractional indices like $x^{\frac{1}{2}}$? These extensions to the index laws allow us to express a wider range of mathematical ideas compactly, from reciprocals to roots of numbers.

🎯 Learning Objectives

By the end of this lesson, you will be able to:

Understand and interpret negative indices as reciprocals
Recognize fractional indices as roots of numbers
Apply the laws of indices to expressions with negative and fractional powers
Calculate numerical values of expressions with negative and fractional indices

🔑 Key Concepts

Negative Indices

A negative index indicates the reciprocal of the base raised to the corresponding positive power.

📖 Definition

Negative Index: For any non-zero number $a$ and positive number $n$:

a^{-n} = \frac{1}{a^n}

This means that when we see a negative index, we take the reciprocal of the base raised to the positive value of that index.

💡 Pro Tip

When working with negative indices, you can first convert them to fractions with positive indices before proceeding with further calculations. This often simplifies the problem.

Fractional Indices

📖 Definition

Fractional Index: For any non-negative number $a$ and positive integers $m$ and $n$:

a^{\frac{1}{n}} = \sqrt[n]{a}$$ $$a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m

This means that:

$a^{\frac{1}{n}}$ represents the $n$th root of $a$
$a^{\frac{m}{n}}$ represents the $n$th root of $a^m$, or the $m$th power of the $n$th root of $a$

Combining Negative and Fractional Indices

We can also have indices that are both negative and fractional. The rules still apply:

a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}} = \frac{1}{\sqrt[n]{a^m}}

📝 Worked Examples

📋 Example 1: Negative Indices

Question: Evaluate $2^{-3}$.

Solution:

Step 1:

Apply the negative index rule: $a^{-n} = \frac{1}{a^n}$

2^{-3} = \frac{1}{2^3}

Step 2:

Calculate $2^3$.

2^3 = 2 \times 2 \times 2 = 8

Final Answer:

Therefore:

2^{-3} = \frac{1}{2^3} = \frac{1}{8} = 0.125

📋 Example 2: Fractional Indices

Question: Evaluate $9^{\frac{1}{2}}$.

Solution:

Step 1:

Apply the fractional index rule: $a^{\frac{1}{n}} = \sqrt[n]{a}$

9^{\frac{1}{2}} = \sqrt{9}

Final Answer:

Therefore:

9^{\frac{1}{2}} = \sqrt{9} = 3

📋 Example 3: Mixed Fractional Indices

Question: Evaluate $8^{\frac{2}{3}}$.

Solution:

Step 1:

Apply the fractional index rule: $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ or $a^{\frac{m}{n}} = (\sqrt[n]{a})^m$

Let's use the second form:

8^{\frac{2}{3}} = (8^{\frac{1}{3}})^2 = (\sqrt[3]{8})^2

Step 2:

Calculate $\sqrt[3]{8}$.

\sqrt[3]{8} = 2

because $2^3 = 8$.

Final Answer:

Therefore:

8^{\frac{2}{3}} = (8^{\frac{1}{3}})^2 = (\sqrt[3]{8})^2 = 2^2 = 4

📋 Example 4: Combined Negative and Fractional Indices

Question: Simplify $4^{-\frac{1}{2}}$.

Solution:

Step 1:

Apply the negative index rule: $a^{-n} = \frac{1}{a^n}$

4^{-\frac{1}{2}} = \frac{1}{4^{\frac{1}{2}}}

Step 2:

Apply the fractional index rule: $a^{\frac{1}{n}} = \sqrt[n]{a}$

4^{\frac{1}{2}} = \sqrt{4} = 2

Final Answer:

Therefore:

4^{-\frac{1}{2}} = \frac{1}{4^{\frac{1}{2}}} = \frac{1}{2} = 0.5

⚠️ Common Mistakes

🚫 Mistake 1: Confusing $a^{-n}$ with $-a^n$

What students often do wrong: Thinking that $a^{-n}$ is the same as $-a^n$.

Why it's wrong: $a^{-n}$ means $\frac{1}{a^n}$, while $-a^n$ means the negative of $a^n$.

How to avoid it: Remember that the negative sign in the exponent means reciprocal, not negative value. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$, but $-2^3 = -(2^3) = -8$.

🚫 Mistake 2: Incorrect Simplification of Fractional Indices

What students often do wrong: Incorrectly applying the rules for fractional indices or trying to use shortcuts.

Why it's wrong: Fractional indices have specific meanings related to roots and powers, and incorrect application can lead to wrong answers.

How to avoid it: Remember that $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$. Always double-check your work and use the correct sequence of operations.

💪 Practice Problems

Try these problems to test your understanding:

🎯 Practice Question 1

Evaluate the following:

(a) $5^{-2}$

(b) $16^{\frac{1}{2}}$

(c) $27^{\frac{2}{3}}$

(d) $25^{-\frac{1}{2}}$

🔍 Show Solution

(a) $5^{-2} = \frac{1}{5^2} = \frac{1}{25} = 0.04$

(b) $16^{\frac{1}{2}} = \sqrt{16} = 4$

(c) $27^{\frac{2}{3}} = (\sqrt[3]{27})^2 = 3^2 = 9$

(d) $25^{-\frac{1}{2}} = \frac{1}{25^{\frac{1}{2}}} = \frac{1}{\sqrt{25}} = \frac{1}{5} = 0.2$

🎯 Practice Question 2

Simplify the following expressions:

(a) $2^3 \times 2^{-5}$

(b) $\frac{3^{-2}}{3^{-4}}$

(c) $(4^{\frac{1}{2}})^3$

(d) $9^{\frac{3}{2}} \div 9^{\frac{1}{2}}$

🔍 Show Solution

(a) $2^3 \times 2^{-5} = 2^{3 + (-5)} = 2^{-2} = \frac{1}{2^2} = \frac{1}{4} = 0.25$

(b) $\frac{3^{-2}}{3^{-4}} = 3^{-2-(-4)} = 3^{-2+4} = 3^2 = 9$

(c) $(4^{\frac{1}{2}})^3 = 4^{\frac{1}{2} \times 3} = 4^{\frac{3}{2}} = (\sqrt{4})^3 = 2^3 = 8$

(d) $9^{\frac{3}{2}} \div 9^{\frac{1}{2}} = 9^{\frac{3}{2} - \frac{1}{2}} = 9^1 = 9$

📋 Summary

🎯 Key Takeaways

Negative indices represent reciprocals: $a^{-n} = \frac{1}{a^n}$
Fractional indices represent roots: $a^{\frac{1}{n}} = \sqrt[n]{a}$
Mixed fractional indices combine both concepts: $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$
The standard laws of indices still apply for negative and fractional indices
Negative fractional indices represent the reciprocal of a root: $a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}} = \frac{1}{\sqrt[n]{a^m}}$

📚 What's Next?

Now that you understand negative and fractional indices, you're ready to learn about surds - irrational numbers expressed using root symbols. Surds are closely related to fractional indices and are essential for exact calculations in advanced mathematics.