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.

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

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

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

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

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

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

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

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

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

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

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

show that the set of all natural number form semi group under the composition of addition
what is the meaning
Dominic
explain and give four Example hyperbolic function
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
felecia
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
×/×+9+6/1
Debbie
Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Naagmenkoma
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
please can go further on polynomials quadratic
Abdullahi
hi mam
Mark
I need quadratic equation link to Alpa Beta
find the value of 2x=32
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
16
Makan
x=16
Makan
use the y -intercept and slope to sketch the graph of the equation y=6x
how do we prove the quadratic formular
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak