# 1.3 Radicals and rational expressions  (Page 5/11)

 Page 5 / 11

## Simplifying rational exponents

Simplify:

1. $5\left(2{x}^{\frac{3}{4}}\right)\left(3{x}^{\frac{1}{5}}\right)$
2. ${\left(\frac{16}{9}\right)}^{-\frac{1}{2}}$

Simplify $\text{\hspace{0.17em}}{\left(8x\right)}^{\frac{1}{3}}\left(14{x}^{\frac{6}{5}}\right).$

$28{x}^{\frac{23}{15}}$

Access these online resources for additional instruction and practice with radicals and rational exponents.

## Key concepts

• The principal square root of a number $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is the nonnegative number that when multiplied by itself equals $\text{\hspace{0.17em}}a.\text{\hspace{0.17em}}$ See [link] .
• If $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ are nonnegative, the square root of the product $\text{\hspace{0.17em}}ab\text{\hspace{0.17em}}$ is equal to the product of the square roots of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ See [link] and [link] .
• If $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ are nonnegative, the square root of the quotient $\text{\hspace{0.17em}}\frac{a}{b}\text{\hspace{0.17em}}$ is equal to the quotient of the square roots of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ See [link] and [link] .
• We can add and subtract radical expressions if they have the same radicand and the same index. See [link] and [link] .
• Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. See [link] and [link] .
• The principal n th root of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is the number with the same sign as $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ that when raised to the n th power equals $\text{\hspace{0.17em}}a.\text{\hspace{0.17em}}$ These roots have the same properties as square roots. See [link] .
• Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals. See [link] and [link] .
• The properties of exponents apply to rational exponents. See [link] .

## Verbal

What does it mean when a radical does not have an index? Is the expression equal to the radicand? Explain.

When there is no index, it is assumed to be 2 or the square root. The expression would only be equal to the radicand if the index were 1.

Where would radicals come in the order of operations? Explain why.

Every number will have two square roots. What is the principal square root?

The principal square root is the nonnegative root of the number.

Can a radical with a negative radicand have a real square root? Why or why not?

## Numeric

For the following exercises, simplify each expression.

$\sqrt{256}$

16

$\sqrt{\sqrt{256}}$

$\sqrt{4\left(9+16\right)}$

10

$\sqrt{289}-\sqrt{121}$

$\sqrt{196}$

14

$\sqrt{1}$

$\sqrt{98}$

$7\sqrt{2}$

$\sqrt{\frac{27}{64}}$

$\sqrt{\frac{81}{5}}$

$\frac{9\sqrt{5}}{5}$

$\sqrt{800}$

$\sqrt{169}+\sqrt{144}$

25

$\sqrt{\frac{8}{50}}$

$\frac{18}{\sqrt{162}}$

$\sqrt{2}$

$\sqrt{192}$

$14\sqrt{6}-6\sqrt{24}$

$2\sqrt{6}$

$15\sqrt{5}+7\sqrt{45}$

$\sqrt{150}$

$5\sqrt{6}$

$\sqrt{\frac{96}{100}}$

$\left(\sqrt{42}\right)\left(\sqrt{30}\right)$

$6\sqrt{35}$

$12\sqrt{3}-4\sqrt{75}$

$\sqrt{\frac{4}{225}}$

$\frac{2}{15}$

$\sqrt{\frac{405}{324}}$

$\sqrt{\frac{360}{361}}$

$\frac{6\sqrt{10}}{19}$

$\frac{5}{1+\sqrt{3}}$

$\frac{8}{1-\sqrt{17}}$

$-\frac{1+\sqrt{17}}{2}$

$\sqrt[4]{16}$

$\sqrt[3]{128}+3\sqrt[3]{2}$

$7\sqrt[3]{2}$

$\sqrt[5]{\frac{-32}{243}}$

$\frac{15\sqrt[4]{125}}{\sqrt[4]{5}}$

$15\sqrt{5}$

$3\sqrt[3]{-432}+\sqrt[3]{16}$

## Algebraic

For the following exercises, simplify each expression.

$\sqrt{400{x}^{4}}$

$20{x}^{2}$

$\sqrt{4{y}^{2}}$

$\sqrt{49p}$

$7\sqrt{p}$

${\left(144{p}^{2}{q}^{6}\right)}^{\frac{1}{2}}$

${m}^{\frac{5}{2}}\sqrt{289}$

$18{m}^{2}\sqrt{m}$

$9\sqrt{3{m}^{2}}+\sqrt{27}$

$3\sqrt{a{b}^{2}}-b\sqrt{a}$

$2b\sqrt{a}$

$\frac{4\sqrt{2n}}{\sqrt{16{n}^{4}}}$

$\sqrt{\frac{225{x}^{3}}{49x}}$

$\frac{15x}{7}$

$3\sqrt{44z}+\sqrt{99z}$

$\sqrt{50{y}^{8}}$

$5{y}^{4}\sqrt{2}$

$\sqrt{490b{c}^{2}}$

$\sqrt{\frac{32}{14d}}$

$\frac{4\sqrt{7d}}{7d}$

${q}^{\frac{3}{2}}\sqrt{63p}$

$\frac{\sqrt{8}}{1-\sqrt{3x}}$

$\frac{2\sqrt{2}+2\sqrt{6x}}{1-3x}$

$\sqrt{\frac{20}{121{d}^{4}}}$

${w}^{\frac{3}{2}}\sqrt{32}-{w}^{\frac{3}{2}}\sqrt{50}$

$-w\sqrt{2w}$

$\sqrt{108{x}^{4}}+\sqrt{27{x}^{4}}$

$\frac{\sqrt{12x}}{2+2\sqrt{3}}$

$\frac{3\sqrt{x}-\sqrt{3x}}{2}$

$\sqrt{147{k}^{3}}$

$\sqrt{125{n}^{10}}$

$5{n}^{5}\sqrt{5}$

$\sqrt{\frac{42q}{36{q}^{3}}}$

$\sqrt{\frac{81m}{361{m}^{2}}}$

$\frac{9\sqrt{m}}{19m}$

$\sqrt{72c}-2\sqrt{2c}$

$\sqrt{\frac{144}{324{d}^{2}}}$

$\frac{2}{3d}$

$\sqrt[3]{24{x}^{6}}+\sqrt[3]{81{x}^{6}}$

$\sqrt[4]{\frac{162{x}^{6}}{16{x}^{4}}}$

$\frac{3\sqrt[4]{2{x}^{2}}}{2}$

$\sqrt[3]{64y}$

$\sqrt[3]{128{z}^{3}}-\sqrt[3]{-16{z}^{3}}$

$6z\sqrt[3]{2}$

$\sqrt[5]{1,024{c}^{10}}$

## Real-world applications

A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So the length of the guy wire can be found by evaluating $\text{\hspace{0.17em}}\sqrt{90,000+160,000}.\text{\hspace{0.17em}}$ What is the length of the guy wire?

500 feet

A car accelerates at a rate of where t is the time in seconds after the car moves from rest. Simplify the expression.

## Extensions

For the following exercises, simplify each expression.

$\frac{\sqrt{8}-\sqrt{16}}{4-\sqrt{2}}-{2}^{\frac{1}{2}}$

$\frac{-5\sqrt{2}-6}{7}$

$\frac{{4}^{\frac{3}{2}}-{16}^{\frac{3}{2}}}{{8}^{\frac{1}{3}}}$

$\frac{\sqrt{m{n}^{3}}}{{a}^{2}\sqrt{{c}^{-3}}}\cdot \frac{{a}^{-7}{n}^{-2}}{\sqrt{{m}^{2}{c}^{4}}}$

$\frac{\sqrt{mnc}}{{a}^{9}cmn}$

$\frac{a}{a-\sqrt{c}}$

$\frac{x\sqrt{64y}+4\sqrt{y}}{\sqrt{128y}}$

$\frac{2\sqrt{2}x+\sqrt{2}}{4}$

$\left(\frac{\sqrt{250{x}^{2}}}{\sqrt{100{b}^{3}}}\right)\left(\frac{7\sqrt{b}}{\sqrt{125x}}\right)$

$\sqrt{\frac{\sqrt[3]{64}+\sqrt[4]{256}}{\sqrt{64}+\sqrt{256}}}$

$\frac{\sqrt{3}}{3}$

#### Questions & Answers

what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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