 Fractional Indices with Algebraic Terms

Lesson

The fractional index rule states:

$x^{\frac{1}{n}}=\sqrt[n]{x}$x1n=nx

Let's look at why this is true using the multiplication law.

Let's say I wanted to solve $25^{\frac{1}{2}}\times25^{\frac{1}{2}}$2512×2512. Using the multiplication law, this would be the same as $25^{\frac{1}{2}+\frac{1}{2}}=25^1$2512+12=251 or $25$25. When we multiply a number by itself, the end product is called a square number. Since we know that $25^{\frac{1}{2}}\times25^{\frac{1}{2}}=25$2512×2512=25, $25^{\frac{1}{2}}$2512 must equal the square root of $25$25, which is $5$5. This must be right as $5\times5=25$5×5=25.

More complicated fractional indices

You may also see questions with more complicated fractional indices, such as $x^{\frac{3}{2}}$x32. We could express this as a power of a power, $\left(x^3\right)^{\frac{1}{2}}$(x3)12. As such, the numerator in the fractional index can be expressed as a power and the denominator in the fractional index can be expressed as a root. More generally, this rule states:

$x^{\frac{m}{n}}=\sqrt[n]{x^m}$xmn=nxm

When solving problems with fractional indices, it doesn't matter whether you start with the powers or the roots (although you might find it easier to do it one way than the other). For example, let's look at $16^{\frac{3}{2}}$1632:

$\sqrt{16^3}=4^3$163=43

$4^3=64$43=64

$\sqrt{16^3}=\sqrt{4096}$163=4096

$\sqrt{4096}=64$4096=64

See, we get the same answer both ways.

Examples

Question 1

Evaluate: $121^{\frac{1}{2}}$12112.

Think: A fractional index of $\frac{1}{2}$12 is the same as finding the square root.

Do:

 $121^{\frac{1}{2}}$12112​ $=$= $\sqrt{121}$√121 $=$= $11$11

Question 2

Express as a fraction in simplest form.

$625^{\frac{-3}{4}}$62534

Question 3

Express $\sqrt{x^7}$5x7 in index form.