# 1.3 Radicals and rational exponents  (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] .
• 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] .
• 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.

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

$20{x}^{2}$

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

$\sqrt{49p}$

$7\sqrt{p}$

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

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

$17{m}^{2}\sqrt{m}$

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

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

$2b\sqrt{a}$

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

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

$\frac{15x}{7}$

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

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

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

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

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

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

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

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

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

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

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

$-w\sqrt{2w}$

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

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

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

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

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

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

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

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

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

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

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

$\frac{2}{3d}$

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

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

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

$\sqrt[3]{64y}$

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

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

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

## 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.

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

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

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

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

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

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

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

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

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

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

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

what are you up to?
nothing up todat yet
Miranda
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jai
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jai
Miranda Drice
jai
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jai
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Miranda
I am living in india
jai
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Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
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Jeffrey
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Miranda
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Jeffrey
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Miranda
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Jeffrey
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Miranda
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Miranda
Jeffrey
Jeffrey
Miranda
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Miranda
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Steve
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Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
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SORIE
Hiiii
Chinni
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Ranjay
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ANSHU
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Chinni
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Chinni
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Hassan
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SORIE
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Abdel
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SORIE
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Yaona
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SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function