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(Video) Simplifying Radical Expressions

by mathman1024

This video by mathman1024 works through converting fractional exponents into radicals.

$\large{\text{Summary}}$

There are many rules about the combination or breakdown of exponents.

$$\begin{align} &a^{\frac{m}{n}}=\sqrt[n]{a}^{\,m}\\ &a^m\cdot a^n=a^{m+n}\\ &\frac{a^m}{a^n}=a^{m-n}\\ &(a^m)^n=a^{m\cdot n} \end{align}$$

To review the laws of exponents, you can check out the lessons on: negative exponents, product rule, quotient rule, and power to a power.

The most relevant rule for this section is the first one with a fractional exponent. The denominator of the exponent is the degree of the radical. The numerator of the exponent is the power of the term inside of the radical.

We can combine this with the other rules to simplify radical and exponential expressions.

The first example is: $\sqrt{x}\cdot\sqrt[5]{x}$. $$\begin{align} \sqrt{x}\cdot\sqrt[5]{x}&=x^{\frac12}\cdot x^{\frac15}\\ &=x^{\frac12+\frac15}\\ &=x^{\frac{7}{10}}\\ &=\sqrt[10]{x}^{\,7} \end{align}$$

The second example he does is: $\dfrac{\sqrt[4]{x^3}}{\sqrt[3]{x^2}}$.

$$\begin{align} \dfrac{\sqrt[4]{x^3}}{\sqrt[3]{x^2}}&=\frac{x^{\frac34}}{x^{\frac23}}\\ &=x^{\frac34-\frac23}\\ &=x^{\frac{1}{12}}\\ &=\sqrt[12]{x} \end{align}$$

These examples will be really helpful when solving word problems!

Fraction exponents are like an exponent and a radical

Fraction exponents tell you to take the power of a number, and then take a root. It's hard to explain in words, so here's a rule:

$$a^{\frac{b}{c}} = \sqrt[c]{a^b} = (\sqrt[c]{a})^b$$

Basically, in a fraction exponent the numerator (top) is the power, and the denominator (bottom) is the root you need to take.

Let's look at a quick example using real numbers: $27^{\frac53}$

Since $5$ is the top of the fraction, we're going to take the expression to the $5$th power.

Since $3$ is the bottom of the fraction, we're going to find the $3$rd root of the expression.

We can do those operations in any order: we can take $27$ to the $5$th power first, then take the $3$rd root ($\sqrt[3]{27^5}$), or we can take the $3$rd root of $27$ and then take it to the $5$th power ($(\sqrt[3]{27})^5$).

Let's do the second option, where we take the root first.

$$3^{\frac53}$$ $$= (\sqrt[3]{27})^5$$ $$= (3)^5$$ $$= 243$$

And the first option:

$$3^{\frac53}$$ $$= \sqrt[3]{27^5}$$ $$= \sqrt[3]{14348907}$$ $$= 243$$

Notice that the numbers in the first version get much bigger much faster. I tend to try to take the root first, if the number makes that easy, but chances are that the numbers won't be pretty.

A rational exponent can always be turned into a radical. If $b$ to the $n^{th}$ root of a number is real, and $m$ is a positive integer, then rational exponents and radicals are related like this: $$b^{\frac{1}{n}}=\sqrt[n]{b}\;\;\text{and}\;\;b^{\frac{m}{n}}=\sqrt[n]{b^m}=(\sqrt[n]{b})^m$$

The denominator of the fraction is the value of the root. For example, if the denominator is $3$, you'll be taking the cube root. You may be dealing with positive or negative fractions, so be aware of that. Positive numbers are most common, as in the example above and below. Negative numbers means you'll need to follow the rules of exponent.

Let's try this with a number first.

Yes, the correct answer is $64$! There are a couple ways we could have gone about this.

First is cubing $16$ first and then taking the square root of that answer like this:

$$16^{\frac32}=\sqrt{16^3}=\sqrt{4096}=64$$

Now, unless you used a calculator, this would be crazy hard to do by hand or in your head. Is there and easier way to do this? Yes there is. You can take the square root of $16$ first and then cube that answer like this:

$$16^{\frac32}=(\sqrt{16})^3=(4)^3=64$$

Much easier. You have both of these options when switching from rational exponents to radicals.