9.7 Higher roots (Page 3/8)

<< Chapter < Page

Elementary algebra Page 3 / 8

Chapter >> Page >

Simplify: ⓐ $\sqrt[3]{−108}$ ⓑ $\sqrt[4]{−48}$ .

ⓐ $−3 \sqrt[3]{4}$ ⓑ $not real$

Got questions? Get instant answers now!

Simplify: ⓐ $\sqrt[3]{−625}$ ⓑ $\sqrt[4]{−324}$ .

ⓐ $−5 \sqrt[3]{5}$ ⓑ $not real$

Got questions? Get instant answers now!

Use the quotient property to simplify expressions with higher roots

We can simplify higher roots with quotients in the same way we simplified square roots. First we simplify any fractions inside the radical.

Simplify: ⓐ $\sqrt[3]{\frac{a^{8}}{a^{5}}}$ ⓑ $\sqrt[4]{\frac{a^{10}}{a^{2}}}$ .

Solution

ⓐ
$\begin{matrix} \sqrt[3]{\frac{a^{8}}{a^{5}}} \\ Simplify the fraction under the radical first. & \sqrt[3]{a^{3}} \\ Simplify. & a \end{matrix}$
ⓑ
$\begin{matrix} \sqrt[4]{\frac{a^{10}}{a^{2}}} \\ Simplify the fraction under the radical first. & \sqrt[4]{a^{8}} \\ Rewrite the radicand using perfect fourth power factors. & \sqrt[4]{{(a^{2})}^{4}} \\ Simplify. & a^{2} \end{matrix}$

Got questions? Get instant answers now!

Simplify: ⓐ $\sqrt[4]{\frac{x^{7}}{x^{3}}}$ ⓑ $\sqrt[4]{\frac{y^{17}}{y^{5}}}$ .

ⓐ $| x |$ ⓑ $y^{3}$

Got questions? Get instant answers now!

Simplify: ⓐ $\sqrt[3]{\frac{m^{13}}{m^{7}}}$ ⓑ $\sqrt[5]{\frac{n^{12}}{n^{2}}}$ .

ⓐ $m^{2}$ ⓑ $n^{2}$

Got questions? Get instant answers now!

Previously, we used the Quotient Property ‘in reverse’ to simplify square roots. Now we will generalize the formula to include higher roots.

Quotient property of n Th roots

\begin{matrix} \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} & and & \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}} \end{matrix}

when $\sqrt[n]{a} and \sqrt[n]{b} are real numbers, b \neq 0, and for any integer n \geq 2$

Simplify: ⓐ $\frac{\sqrt[3]{−108}}{\sqrt[3]{2}}$ ⓑ $\frac{\sqrt[4]{96 x^{7}}}{\sqrt[4]{3 x^{2}}}$ .

Solution

ⓐ
$\begin{matrix} \frac{\sqrt[3]{−108}}{\sqrt[3]{2}} \\ \begin{matrix} Neither radicand is a perfect cube, so use \\ the Quotient Property to write as one radical. \end{matrix} & \sqrt[3]{\frac{−108}{2}} \\ Simplify the fraction under the radical. & \sqrt[3]{−54} \\ \begin{matrix} Rewrite the radicand as a product using \\ perfect cube factors. \end{matrix} & \sqrt[3]{{(−3)}^{3} \cdot 2} \\ Rewrite the radical as the product of two radicals. & \sqrt[3]{{(−3)}^{3}} \cdot \sqrt[3]{2} \\ Simplify. & −3 \sqrt[3]{2} \end{matrix}$
ⓑ
$\begin{matrix} \frac{\sqrt[4]{96 x^{7}}}{\sqrt[4]{3 x^{2}}} \\ \begin{matrix} Neither radicand is a perfect fourth power, \\ so use the Quotient Property to write as one radical. \end{matrix} & \sqrt[4]{\frac{96 x^{7}}{3 x^{2}}} \\ Simplify the fraction under the radical. & \sqrt[4]{32 x^{5}} \\ \begin{matrix} Rewrite the radicand as a product using \\ perfect fourth power factors. \end{matrix} & \sqrt[4]{2^{4} x^{4} \cdot 2 x} \\ Rewrite the radical as the product of two radicals. & \sqrt[4]{{(2 x)}^{4}} \cdot \sqrt[4]{2 x} \\ Simplify. & 2 | x | \sqrt[4]{2 x} \end{matrix}$

Got questions? Get instant answers now!

Simplify: ⓐ $\frac{\sqrt[3]{−532}}{\sqrt[3]{2}}$ ⓑ $\frac{\sqrt[4]{486 m^{11}}}{\sqrt[4]{3 m^{5}}}$ .

ⓐ $not real$ ⓑ $3 | m | \sqrt[4]{2 m^{2}}$

Got questions? Get instant answers now!

Simplify: ⓐ $\frac{\sqrt[3]{−192}}{\sqrt[3]{3}}$ ⓑ $\frac{\sqrt[4]{324 n^{7}}}{\sqrt[4]{2 n^{3}}}$ .

ⓐ $−4$ ⓑ $3 | n | \sqrt[4]{2}$

Got questions? Get instant answers now!

If the fraction inside the radical cannot be simplified, we use the first form of the Quotient Property to rewrite the expression as the quotient of two radicals.

Simplify: ⓐ $\sqrt[3]{\frac{24 x^{7}}{y^{3}}}$ ⓑ $\sqrt[4]{\frac{48 x^{10}}{y^{8}}}$ .

Solution

ⓐ
$\begin{matrix} \sqrt[3]{\frac{24 x^{7}}{y^{3}}} \\ \begin{matrix} The fraction in the radicand cannot be \\ simplified. Use the Quotient Property to \\ write as two radicals. \end{matrix} & \frac{\sqrt[3]{24 x^{7}}}{\sqrt[3]{y^{3}}} \\ \begin{matrix} Rewrite each radicand as a product using \\ perfect cube factors. \end{matrix} & \frac{\sqrt[3]{8 x^{6} \cdot 3 x}}{\sqrt[3]{y^{3}}} \\ Rewrite the numerator as the product of two radicals. & \frac{\sqrt[3]{{(2 x^{2})}^{3}} \sqrt[3]{3 x}}{\sqrt[3]{y^{3}}} \\ Simplify. & \frac{2 x^{2} \sqrt[3]{3 x}}{y} \end{matrix}$
ⓑ
$\begin{matrix} \sqrt[4]{\frac{48 x^{10}}{y^{8}}} \\ \begin{matrix} The fraction in the radicand cannot be \\ simplified. Use the Quotient Property to \\ write as two radicals. \end{matrix} & \frac{\sqrt[4]{48 x^{10}}}{\sqrt[4]{y^{8}}} \\ \begin{matrix} Rewrite each radicand as a product using \\ perfect fourth power factors. \end{matrix} & \frac{\sqrt[4]{16 x^{8} \cdot 3 x^{2}}}{\sqrt[4]{y^{8}}} \\ Rewrite the numerator as the product of two radicals. & \frac{\sqrt[4]{{(2 x^{2})}^{4}} \sqrt[4]{3 x^{2}}}{\sqrt[4]{{(y^{2})}^{4}}} \\ Simplify. & \frac{2 x^{2} \sqrt[4]{3 x^{2}}}{y^{2}} \end{matrix}$

Got questions? Get instant answers now!

Simplify: ⓐ $\sqrt[3]{\frac{108 c^{10}}{d^{6}}}$ ⓑ $\sqrt[4]{\frac{80 x^{10}}{y^{5}}}$ .

ⓐ $\frac{3 c^{3} \sqrt[3]{4 c}}{d^{2}}$ ⓑ $\frac{x^{2}}{| y |} \sqrt[4]{\frac{80 x^{2}}{y}}$

Got questions? Get instant answers now!

Simplify: ⓐ $\sqrt[3]{\frac{40 r^{3}}{s}}$ ⓑ $\sqrt[4]{\frac{162 m^{14}}{n^{12}}}$ .

ⓐ $r \sqrt[3]{\frac{40}{s}}$ ⓑ $\frac{3 m^{3} \sqrt[4]{2 m^{2}}}{| n^{3} |}$

Got questions? Get instant answers now!

Add and subtract higher roots

We can add and subtract higher roots like we added and subtracted square roots. First we provide a formal definition of like radicals .

Like radicals

Radicals with the same index and same radicand are called like radicals .

Like radicals have the same index and the same radicand.

$9 \sqrt[4]{42 x}$ and $−2 \sqrt[4]{42 x}$ are like radicals.
$5 \sqrt[3]{125 x}$ and $6 \sqrt[3]{125 y}$ are not like radicals. The radicands are different.
$2 \sqrt[5]{1000 q}$ and $−4 \sqrt[4]{1000 q}$ are not like radicals. The indices are different.

We add and subtract like radicals in the same way we add and subtract like terms. We can add $9 \sqrt[4]{42 x} + (−2 \sqrt[4]{42 x})$ and the result is $7 \sqrt[4]{42 x}$ .

Simplify: ⓐ $\sqrt[3]{4 x} + \sqrt[3]{4 x}$ ⓑ $4 \sqrt[4]{8} - 2 \sqrt[4]{8}$ .

Solution

ⓐ
$\begin{matrix} \sqrt[3]{4 x} + \sqrt[3]{4 x} \\ The radicals are like, so we add the coefficients. & 2 \sqrt[3]{4 x} \end{matrix}$
ⓑ
$\begin{matrix} 4 \sqrt[4]{8} - 2 \sqrt[4]{8} \\ The radicals are like, so we subtract the coefficients. & 2 \sqrt[4]{8} \end{matrix}$

Got questions? Get instant answers now!

Questions & Answers

what is biology

Hajah Reply

the study of living organisms and their interactions with one another and their environments

AI-Robot

what is biology

Victoria Reply

HOW CAN MAN ORGAN FUNCTION

Alfred Reply

the diagram of the digestive system

Assiatu Reply

allimentary cannel

Ogenrwot

How does twins formed

William Reply

They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together

Oluwatobi

what is genetics

Josephine Reply

Genetics is the study of heredity

Misack

how does twins formed?

Misack

What is manual

Hassan Reply

discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles

Joseph Reply

what is biology

Yousuf Reply

the study of living organisms and their interactions with one another and their environment.

Wine

discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form

Joseph Reply

what is the blood cells

Shaker Reply

list any five characteristics of the blood cells

Shaker

lack electricity and its more savely than electronic microscope because its naturally by using of light

Abdullahi Reply

advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear

Abdullahi

cell theory state that every organisms composed of one or more cell,cell is the basic unit of life

Abdullahi

is like gone fail us

DENG

cells is the basic structure and functions of all living things

Ramadan

What is classification

ISCONT Reply

is organisms that are similar into groups called tara

Yamosa

in what situation (s) would be the use of a scanning electron microscope be ideal and why?

Kenna Reply

A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,

Hilary

cell is the building block of life.

Condoleezza Reply

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 4

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask

	10 Neuroanatomy 10 Corticospinal Tract By Stephen Voron Start Quiz
	College physics By OpenStax Read Online Course
	My OCA Mock By Mike Wolf Start Exam
©flickr: Abraham	2 Biology 2 By Sarah Warren Start Test
	12 AP 12 Nervous System Essay By OpenStax Start Flashcards
	Western Political Thought MCQ By Saylor Foundation Start Quiz
	6 Psychology MCQ 2010 2 Exam By John Gabrieli Start Exam
	3 Neuroscience Exam 2004 4 By David Corey Start Exam
	Social Dances 1 By Marion Cabalfin Start Test
	Concepts of biology By OpenStax Read Online Course