Rational Indices Revision Notes for VCE SSCE Mathematical Methods

Rational Indices

Understanding rational indices

When working with powers, we're not limited to whole number exponents. We can also use fractions as indices, which are called rational indices. These fractional powers have a special relationship with roots.

For any positive real number $a$ and natural number $n$ , we define:

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

This means that $a^{\frac{1}{n}}$ is the positive number that, when raised to the power of n, gives us a. We can express this relationship as:

(a^{\frac{1}{n}})^n = a

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For example, $9^{\frac{1}{2}} = 3$ because $3^2 = 9$ . The fractional index $\frac{1}{2}$ tells us to find the square root.

Square root notation

We can write square roots using this fractional index notation:

\sqrt{a} = \sqrt[2]{a} = a^{\frac{1}{2}}

This shows that taking the square root is the same as raising to the power of $\frac{1}{2}$ .

Special cases with rational indices

Zero raised to fractional powers

For any natural number $n$ :

0^{\frac{1}{n}} = 0

This works because $0^n = 0$ for any positive integer $n$ .

Negative bases with odd roots

When $n$ is an odd number, we can extend the definition of $a^{\frac{1}{n}}$ to negative values of $a$ . If $a$ is negative and $n$ is odd, then $a^{\frac{1}{n}}$ is the number whose $n$ th power equals $a$ .

chatImportant

Remember: Odd roots welcome negatives!

Odd roots can handle negative bases: $(-8)^{\frac{1}{3}} = -2$ because $(-2)^3 = -8$

This only works when the root (the denominator) is odd. Even roots of negative numbers are not defined in the real number system.

General rational indices

We can extend this notation to any rational index of the form $\frac{m}{n}$ , where $m$ and $n$ are integers. The key definition is:

a^{\frac{m}{n}} = (a^{\frac{1}{n}})^m

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Denominator = Root, Numerator = Power

This tells us to:

First find the nth root of a (using the denominator)
Then raise that result to the power of m (using the numerator)

It's important to always write the fractional power in its simplest form before using this definition.

Worked example: Evaluating expressions with rational indices

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Worked Example: Evaluating Rational Indices

a) $(-64)^{\frac{1}{3}}$

(-64)^{\frac{1}{3}} = -4

Since $(-4)^3 = -64$ , the cube root of $-64$ is -4.

b) $9^{-\frac{1}{2}}$

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

A negative index means we take the reciprocal.

c) $16^{\frac{5}{2}}$

16^{\frac{5}{2}} = (16^{\frac{1}{2}})^5 = (\sqrt[2]{16})^5 = 4^5 = 1024

First find the square root, then raise to the power of 5.

d) $64^{-\frac{2}{3}}$

64^{-\frac{2}{3}} = \frac{1}{64^{\frac{2}{3}}} = \frac{1}{(64^{\frac{1}{3}})^2} = \frac{1}{(\sqrt[3]{64})^2} = \frac{1}{4^2} = \frac{1}{16}

The negative index gives us the reciprocal, then we find the cube root and square it.

Index laws for rational indices

The familiar index laws that work for integer powers also apply to rational indices:

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Index Laws for Rational Indices

Law 1: Multiplication

a^{\frac{m}{q}} \times a^{\frac{n}{p}} = a^{\frac{m}{q}+\frac{n}{p}}

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

Law 2: Division

a^{\frac{m}{q}} \div a^{\frac{n}{p}} = a^{\frac{m}{q}-\frac{n}{p}}

When dividing powers with the same base, subtract the indices.

Law 3: Power of a power

(a^{\frac{m}{q}})^{\frac{n}{p}} = a^{\frac{m}{q} \times \frac{n}{p}}

When raising a power to another power, multiply the indices.

These laws are essential for simplifying expressions involving rational indices.

Worked example: Simplifying expressions

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Worked Example: Simplifying Complex Expressions

a) Simplify: $\frac{3^{\frac{1}{4}} \times \sqrt{6} \times \sqrt[4]{2}}{16^{\frac{3}{4}}}$

Step 1: Express all terms using fractional indices:

\sqrt{6} = \sqrt{3} \times \sqrt{2} = 3^{\frac{1}{2}} \times 2^{\frac{1}{2}}

and

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

Step 2: Rewrite the expression:

\frac{3^{\frac{1}{4}} \times \sqrt{6} \times \sqrt[4]{2}}{16^{\frac{3}{4}}} = \frac{3^{\frac{1}{4}} \times 3^{\frac{1}{2}} \times 2^{\frac{1}{2}} \times 2^{\frac{1}{4}}}{(16^{\frac{1}{4}})^3}

Step 3: Simplify the denominator:

(16^{\frac{1}{4}})^3 = 2^3 = 2^{\frac{12}{4}}

= \frac{3^{\frac{1}{4}} \times 3^{\frac{1}{2}} \times 2^{\frac{1}{2}} \times 2^{\frac{1}{4}}}{2^3}

Step 4: Combine powers of the same base:

= \frac{3^{\frac{3}{4}} \times 2^{\frac{3}{4}}}{2^3}

Step 5: Note that $2^3 = 2^{\frac{12}{4}}$ and simplify:

= \frac{3^{\frac{3}{4}}}{2^{\frac{12}{4}-\frac{3}{4}}} = \frac{3^{\frac{3}{4}}}{2^{\frac{9}{4}}}

b) Simplify: $(x^{-2}y)^{\frac{1}{2}} \times (\frac{x}{y^{-3}})^4$

Step 1: Apply the power to each factor:

(x^{-2}y)^{\frac{1}{2}} \times (\frac{x}{y^{-3}})^4 = x^{-1}y^{\frac{1}{2}} \times \frac{x^4}{y^{-12}}

Step 2: Remember that $\frac{x^4}{y^{-12}} = x^4 \times y^{12}$ :

= x^{-1}y^{\frac{1}{2}} \times x^4 \times y^{12}

Step 3: Combine powers of the same base:

= x^3 \times y^{\frac{25}{2}}

Remember!

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Key Points to Remember:

Rational indices connect fractional powers to roots: $a^{\frac{1}{n}} = \sqrt[n]{a}$
For any rational index $\frac{m}{n}$ , we have $a^{\frac{m}{n}} = (a^{\frac{1}{n}})^m$ - take the root first, then raise to the power
Negative bases only work with odd roots (e.g., $(-8)^{\frac{1}{3}} = -2$ )
All three index laws extend to rational indices, making simplification possible
Always express fractional powers in simplest form before evaluating

Rational Indices (VCE SSCE Mathematical Methods): Revision Notes