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### Course: Digital SAT Math>Unit 8

Lesson 2: Radicals and rational exponents: medium

# Radicals and rational exponents | Lesson

A guide to radicals and rational exponents on the digital SAT

## What are radicals and rational exponents?

Exponential expressions are algebraic expressions with a coefficient, one or more variables, and one or more exponents. For example, in the expression $3{x}^{4}$:
• $3$ is the coefficient.
• $x$ is the base.
• $4$ is the exponent.
In $3{x}^{4}$, $3$ is multiplied by $x$ $4$ times:
$3{x}^{4}=3\cdot \left(x\cdot x\cdot x\cdot x\right)$
An expression can also be raised to an exponent. For example, for $\left(3x{\right)}^{4}$, the expression $3x$ is multiplied by itself $4$ times:
$\left(3x{\right)}^{4}=3x\cdot 3x\cdot 3x\cdot 3x=81{x}^{4}$
Notice how $3{x}^{4}\ne \left(3x{\right)}^{4}$ !
Rational exponents refer to exponents that are/can be represented as fractions: $\frac{1}{2}$, $3$, and $-\frac{2}{3}$ are all considered rational exponents. Radicals are another way to write rational exponents. For example, ${x}^{{}^{\frac{1}{2}}}$ and $\sqrt{x}$ are equivalent.
In this lesson, we'll:
1. Review the rules of exponent operations with integer exponents
2. Apply the rules of exponent operations to rational exponents
3. Make connections between equivalent rational and radical expressions
You can learn anything. Let's do this!

## What are the rules of exponent operations?

### Powers of products & quotients (integer exponents)

Powers of products & quotients (integer exponents)See video transcript

### The rules of exponent operations

#### Adding and subtracting exponential expressions

When adding and subtracting exponential expressions, we're essentially combining like terms. That means we can only combine exponential expressions with both the same base and the same exponent.
$\begin{array}{rl}a{x}^{n}±b{x}^{n}& =\left(a±b\right){x}^{n}\end{array}$

#### Multiplying and dividing exponential expressions

When multiplying two exponential expressions with the same base, we keep the base the same, multiply the coefficients, and add the exponents. Similarly, when dividing two exponential expressions with the same base, we keep the base the same and subtract the exponents.
$\begin{array}{rl}a{x}^{m}\cdot b{x}^{n}& =ab\cdot {x}^{m+n}\\ \\ \frac{a{x}^{m}}{b{x}^{n}}& =\frac{a}{b}\cdot {x}^{m-n}\end{array}$
When multiplying or dividing exponential expressions with the same exponent but different bases, we multiply or divide the bases and keep the exponents the same.
$\begin{array}{rl}{x}^{n}\cdot {y}^{n}& =\left(xy{\right)}^{n}\\ \\ \frac{{x}^{n}}{{y}^{n}}& ={\left(\frac{x}{y}\right)}^{n}\end{array}$

#### Raising an exponential expression to an exponent and change of base

When raising an exponential expression to an exponent, raise the coefficient of the expression to the exponent, keep the base the same, and multiply the two exponents.
${\left(a{x}^{m}\right)}^{n}={a}^{n}\cdot {x}^{mn}$
When the bases are numbers, we can use a similar rule to change the base of an exponential expression.
$\left({a}^{b}{\right)}^{n}={a}^{bn}$
This is useful for questions with multiple terms that need to be written in the same base.

#### Negative exponents

A base raised to a negative exponent is equivalent to $1$ divided by the base raised to the
of the exponent.
${x}^{-n}=\frac{1}{{x}^{n}}$

#### Zero exponent

A nonzero base raised to an exponent of $0$ is equal to $1$.
${x}^{0}=1,x\ne 0$

### How do the rules of exponent operations apply to rational exponents?

Every rule that applies to integer exponents also applies to rational exponents.

### Try it!

try: divide two rational expressions
In order to divide $12{x}^{{}^{\frac{5}{2}}}$ by $3{x}^{{}^{\frac{1}{2}}}$, we
the coefficients and
the exponents of $x$.
$\frac{12{x}^{{}^{\frac{5}{2}}}}{3{x}^{{}^{\frac{1}{2}}}}=\phantom{\rule{0.167em}{0ex}}$

Try: raise an exponential expression to an exponent
To calculate ${\left(2{y}^{{}^{\frac{4}{3}}}\right)}^{3}$, we
and
the exponents $\frac{4}{3}$ and $3$.
${\left(2{y}^{{}^{\frac{4}{3}}}\right)}^{3}=\phantom{\rule{0.167em}{0ex}}$

## How are radicals and fractional exponents related?

### Rewriting roots as rational exponents

Rewriting roots as rational exponentsSee video transcript

### Roots and rational exponents

Squares and square roots are inverse operations: they "undo" each other. For example, if we take the square root of $3$ squared, we get $\sqrt{{3}^{2}}=3$.
The reason for this becomes more apparent when we rewrite square root as a fractional exponent: $\sqrt{x}={x}^{{}^{\frac{1}{2}}}$, and $\sqrt{{3}^{2}}=\left({3}^{2}{\right)}^{{}^{\frac{1}{2}}}={3}^{1}$.
When rewriting a radical expression as a fractional exponent, any exponent under the radical symbol ($\sqrt{\phantom{x}}$) becomes the numerator of the fractional exponent, and the value to the left of the radical symbol (e.g., $\sqrt[3]{\phantom{A}\phantom{x}}$) becomes the denominator of the fractional exponent. Square root is equivalent to $\sqrt[2]{\phantom{A}\phantom{x}}$.
$\sqrt[n]{\phantom{A}{x}^{m}}={x}^{{}^{\frac{m}{n}}}$
All of the rules that apply to exponential expressions with integer exponents also apply to exponential expressions with fractional exponents. Similarly, for radical expressions:
$\begin{array}{rl}\sqrt[n]{\phantom{A}x}\cdot \sqrt[n]{\phantom{A}y}& =\sqrt[n]{\phantom{A}xy}\\ \\ \frac{\sqrt[n]{\phantom{A}x}}{\sqrt[n]{\phantom{A}y}}& =\sqrt[n]{\phantom{A}\frac{x}{y}}\end{array}$
When working with radical expressions with the same radical, we can choose whether to convert to fractional exponents first or multiply what's under the radical symbol first to our advantage.

### Try it!

Try: determine equivalent expressions
Determine whether each of the radical expressions below is equivalent to ${x}^{{}^{\frac{3}{2}}}{y}^{{}^{\frac{1}{3}}}$.
Equivalent $\sqrt{{x}^{3}}\cdot \sqrt[3]{\phantom{A}y}$‍ $\sqrt{xy}$‍ $\sqrt{{x}^{3}y}$‍ $\sqrt[6]{\phantom{A}{x}^{9}{y}^{2}}$‍

Practice: multiply rational expressions
Which of the following is equivalent to $2{x}^{3}\cdot 3{x}^{5}$ ?

Practice: change bases
If ${a}^{{}^{\frac{b}{2}}}=25$ for positive integers $a$ and $b$, what is one possible value of $b$ ?

Practice: raise to a negative exponent
If ${n}^{{}^{-\frac{1}{3}}}=x$, where $n>0$, what is $n$ in terms of $x$ ?

$\frac{\sqrt[3]{\phantom{A}8{x}^{8}{y}^{6}}}{\sqrt{4{x}^{2}{y}^{6}}}$
Which of the following is equivalent to the expression above?

## Things to remember

$\begin{array}{rl}a{x}^{n}±b{x}^{n}& =\left(a±b\right){x}^{n}\end{array}$
Multiplying and dividing exponential expressions:
$\begin{array}{rl}a{x}^{m}\cdot b{x}^{n}& =ab\cdot {x}^{m+n}\\ \\ \frac{a{x}^{m}}{b{x}^{n}}& =\frac{a}{b}\cdot {x}^{m-n}\\ \\ {x}^{n}\cdot {y}^{n}& =\left(xy{\right)}^{n}\\ \\ \frac{{x}^{n}}{{y}^{n}}& ={\left(\frac{x}{y}\right)}^{n}\end{array}$
Raising an exponential expression to an exponent and change of base:
$\begin{array}{rl}{\left(a{x}^{m}\right)}^{n}& ={a}^{n}\cdot {x}^{mn}\\ \\ \left({a}^{b}{\right)}^{n}& ={a}^{bn}\end{array}$
Negative exponent:
${x}^{-n}=\frac{1}{{x}^{n}}$
Zero exponent:
${x}^{0}=1,x\ne 0$
All of the rules that apply to exponential expressions with integer exponents also apply to exponential expressions with fractional exponents.
$\begin{array}{rl}\sqrt[n]{\phantom{A}{x}^{m}}& ={x}^{{}^{\frac{m}{n}}}\\ \\ \sqrt[n]{\phantom{A}x}\cdot \sqrt[n]{\phantom{A}y}& =\sqrt[n]{\phantom{A}xy}\\ \\ \frac{\sqrt[n]{\phantom{A}x}}{\sqrt[n]{\phantom{A}y}}& =\sqrt[n]{\phantom{A}\frac{x}{y}}\end{array}$

## Want to join the conversation?

• since i started studying for sat, memories of middle school are coming back
• i clearly remember taking those in grade 8 and i was a master at them but somehow studying for the sat rn just gave me a flashback and im (relearning them now):(
• I feel like giving up
• comeon man you've come too far too give up now keep trying!
• Never back down never WHAT??..
• dumb ways to die
• its easy man, in asia its in the syllabus of 5th or 6th
• I have no idea what's going on in the last question.
• hard for me, lots of things to keep track of. I'll keep trying though
• the hardest part of the sat is actually remembering the things you learned in middle school lmao
• Exactly that
• my DSAT is in august any tips ? please help your sister out its my first time taking any sat and im rlly nervous. i took a practice test and got 1050 any tips?
• Just pick the CORRECT answers on English, not the seem-like-correct ones. For math, just practice on what you are not sure before the test
(1 vote)
• last question is the hardest and has the worst explanation imaginable
• That explanation was really terrible. Here's how I did the problem. I don't know if this explanation will make any sense over writing but I'll try.

First you should distribute the cube root on the top and the square root on the bottom. A cube root is the same as an exponent of 1/3, and a square root is the same as an exponent of 1/2. So you can get rid of them by raising everything in the top row to the 1/3 power and everything in the bottom row to the 1/2 power. The equation now looks like this:
On top: 8^(1/3) · x^(8/3) · y^(6/3)
On bottom: 4^(1/2) · x^(2/2) · y^(6/2)

Now you can simplify some of the fraction exponents just like you'd simplify any fraction:
On top: 8^(1/3) · x^(8/3) · y^(2)
On bottom: 4^(1/2) · x^(1) · y^(3)

8^(1/3) is the same as the cube root of 8. That equals 2. If you didn't know that, you can plug 8^(1/3) into your calculator and you'll get an answer.
4^(1/2) is the same as the square root of 4. That's 2. So now we have this:
On top: 2 · x^(8/3) · y(2)
On bottom: 2 · x^(1) · y^(3)

Now you can simplify each of the variables in the problem. You do this by dividing the top by the bottom.
2 divided by 2 is 1, so the 2's cancel out.
To divide numbers with exponents, you have to subtract the exponents. So for x, you'd do 8/3 minus 1. This is just like subtracting any other fraction: You need to get a common denominator. So 1 becomes 3/3. 8/3 minus 3/3 equals 5/3. So now you have x^(5/3) on the top.
To simplify y, you subtract: 2 minus 3. That equals -1. So you have y^1 on the bottom, or just y.
On top: x^(5/3)
On bottom: y

Now you need to simplify the x^(5/3). (I'm not sure if Khan Academy has taught this in this course yet, but the process I'm using is called "simplify radicals." This is the same as the cube root of (x^5). Inside the radical you can write five 'x's because that's what x^5 is: x·x·x·x·x. Now, because it's a cube root on the outside, you can circle groups of three 'x's from inside. That gives you one group of three, as well as two 'x's that don't fit in the group. Now you can take out the group. You do this by getting rid of all three 'x's from the inside and putting one of them outside the radical, in front of it. The other two 'x's (that didn't fit in a group) stay inside the radical.
On top: x ∛(xx)
On bottom: y

x times x is x^2. So you can simplify the inside of the radical and get the answer:
On top: x · ∛(x^2)
On bottom: y