01Cfractionalindices


 * = Fractional Indices =

We know that a square root multiplied by itself simplifies to the value inside the root: math . \qquad \sqrt{9} \times \sqrt{9} = 9 math

This raises the question: Is there a number in index form that does the same thing? math . \qquad 9^a \times 9^a = 9^1 math

Using the 1st index law {when multiplying numbers with the same base, add the indices} . math \\ . \qquad 9^a \times 9^a = 9^1 \\. \\ . \qquad 9^{2a} = 9^1 \qquad \{ \textit{the indices must be equal} \} \\. \\ . \qquad 2a = 1 \\. \\ . \qquad a = \dfrac{1}{2} math

Therefore: math . \qquad 9^{\frac{1}{2}} \times 9^{\frac{1}{2}} = 9^1 math

Compare this to: math . \qquad \sqrt{9} \times \sqrt{9} = 9 math

and we see that: math . \qquad \sqrt{9} = 9^{\frac{1}{2}} math

In the same way, we know that if we cube a cube root, we get the value inside the cube root: math . \qquad \sqrt[3]{8} \times \sqrt[3]{8} \times \sqrt[3]{8} = 8 math

And we know that using the 1st index law: math . \qquad 8^{\frac{1}{3}} \times 8^{\frac{1}{3}} \times 8^{\frac{1}{3}} = 8^1 math

Therefore: math . \qquad \sqrt[3]{8} = 8^{\frac{1}{3}} math

This leads us to the 8th index law:


 * 8th Index Law (Fractional Indices) **

... When the __**index**__ of a is the fraction 1/n, the number is equivalent to the nth root of a: math . \qquad \cdot \quad a^{\frac{1}{n}} = \sqrt[n]{a} math

Combining this with the 4th index law {raising a number in index form to another power} we get: math . \qquad \qquad a^{\frac{m}{n}} = \Big( a^{\frac{1}{n}} \Big)^m = \Big( \sqrt[n]{a} \Big)^m math or math . \qquad \qquad a^{\frac{m}{n}} = \Big( a^m \Big)^{\frac{1}{n}} = \sqrt[n]{ a^m } math


 * Note: **
 * All of the previous index laws apply to numbers with fractional indices.


 * Example 1 **

... Evaluate the following without using a calculator.

... ... ** 1) ** math \\ . \qquad \qquad 16^{\frac{1}{2}} \\ . \\ . \qquad \qquad = \sqrt{16} \\ . \\ . \qquad \qquad = 4 math

... ... ** 2) ** . math \\ . \qquad \qquad 27^{\frac{4}{3}} \\ . \\ . \qquad \qquad = \Big( \sqrt[3]{27} \Big)^4 \\ . \\ . \qquad \qquad = \Big( 3 \Big)^4 \\ . \\ . \qquad \qquad = 81 math


 * Example 2 **

... Simplify the following

... ... ** 1) ** . math \\ . \qquad \qquad a^{\frac{2}{5}} \times a^{\frac{1}{5}} \\ . \\ . \qquad \qquad = a^{\frac{3}{5}} math

... ... ** 2) ** . math \\ . \qquad \qquad \left( \dfrac{ a^3 } {b^6 } \right) ^{\frac{1}{2} } \\ . \\ . \qquad \qquad = \dfrac{ a^\frac{3}{2} }{ b^3} math

.

.