



Simplifying rational exponents
Simplify:

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

${\left(\frac{16}{9}\right)}^{\frac{1}{2}}$

$\begin{array}{cc}30{x}^{\frac{3}{4}}{x}^{\frac{1}{5}}\hfill & \phantom{\rule{2.5em}{0ex}}\text{Multiplythecoefficients}.\hfill \\ 30{x}^{\frac{3}{4}+\frac{1}{5}}\hfill & \phantom{\rule{2.5em}{0ex}}\text{Usepropertiesofexponents}.\hfill \\ 30{x}^{\frac{19}{20}}\hfill & \phantom{\rule{2.5em}{0ex}}\text{Simplify}.\hfill \end{array}$

$\begin{array}{cc}{\left(\frac{9}{16}\right)}^{\frac{1}{2}}\hfill & \phantom{\rule{2em}{0ex}}\text{\hspace{1em}\hspace{1em}}\text{Usedefinitionofnegativeexponents}.\hfill \\ \sqrt{\frac{9}{16}}\hfill & \phantom{\rule{2em}{0ex}}\text{\hspace{1em}\hspace{1em}}\text{Rewriteasaradical}.\hfill \\ \frac{\sqrt{9}}{\sqrt{16}}\hfill & \phantom{\rule{2em}{0ex}}\text{\hspace{1em}\hspace{1em}}\text{Usethequotientrule}.\hfill \\ \frac{3}{4}\hfill & \phantom{\rule{2em}{0ex}}\text{\hspace{1em}\hspace{1em}}\text{Simplify}.\hfill \end{array}$
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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] .
Section exercises
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.
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Numeric
For the following exercises, simplify each expression.
Algebraic
For the following exercises, simplify each expression.
Realworld 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
$\text{\hspace{0.17em}}\sqrt{\mathrm{90,000}+\mathrm{160,000}}.\text{\hspace{0.17em}}$ What is the length of the guy wire?
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A car accelerates at a rate of
$\text{\hspace{0.17em}}6\frac{\sqrt{4}}{\sqrt{t}}{\text{m/s}}^{2}\text{\hspace{0.17em}}$ where
t is the time in seconds after the car moves from rest. Simplify the expression.
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Extensions
For the following exercises, simplify each expression.
$\frac{\sqrt{m{n}^{3}}}{{a}^{2}\sqrt{{c}^{\mathrm{3}}}}\cdot \frac{{a}^{\mathrm{7}}{n}^{\mathrm{2}}}{\sqrt{{m}^{2}{c}^{4}}}$
$\frac{\sqrt{mnc}}{{a}^{9}cmn}$
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Questions & Answers
nothing up todat yet
Miranda
aap konsi country se ho
jai
which language is that
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
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
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math.
I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
what is the solution of the given equation?
please where is the equation
Miranda
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
and I will walk you through it
Chris
cos (z)= cos z .
Swadesh
what is the identity of 1cos²5x equal to?
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
what is the function of sine with respect of cosine , graphically
sinx sin2x is linearly dependent
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
Source:
OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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