Fractional exponents with algebraic terms

The fractional exponent rule states:

$x^{\frac{1}{n}}=\sqrt[n]{x}$x1n​=ⁿ√x

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 exponents

You may also see questions with more complicated fractional exponents, 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 exponent can be expressed as a power and the denominator in the fractional exponent can be expressed as a root.

More generally, this rule states:

$x^{\frac{m}{n}}=\sqrt[n]{x^m}$xmn​=ⁿ√xm

When solving problems with fractional exponents, 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

Now let's start with the powers:

$\sqrt{16^3}=\sqrt{4096}$√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 exponent 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

Question 3