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The product rule for simplifying square roots

If a and b are nonnegative, the square root of the product a b is equal to the product of the square roots of a and b .

a b = a b

Given a square root radical expression, use the product rule to simplify it.

  1. Factor any perfect squares from the radicand.
  2. Write the radical expression as a product of radical expressions.
  3. Simplify.

Using the product rule to simplify square roots

Simplify the radical expression.

  1. 300
  2. 162 a 5 b 4

  1. 100 3 Factor perfect square from radicand . 100 3 Write radical expression as product of radical expressions . 10 3 Simplify .

  2. 81 a 4 b 4 2 a Factor perfect square from radicand . 81 a 4 b 4 2 a Write radical expression as product of radical expressions . 9 a 2 b 2 2 a Simplify .
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Simplify 50 x 2 y 3 z .

5 | x | | y | 2 y z . Notice the absolute value signs around x and y ? That’s because their value must be positive!

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Given the product of multiple radical expressions, use the product rule to combine them into one radical expression.

  1. Express the product of multiple radical expressions as a single radical expression.
  2. Simplify.

Using the product rule to simplify the product of multiple square roots

Simplify the radical expression.
12 3

12 3 Express the product as a single radical expression . 36 Simplify . 6

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Simplify 50 x 2 x assuming x > 0.

10 | x |

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Using the quotient rule to simplify square roots

Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewrite 5 2 as 5 2 .

The quotient rule for simplifying square roots

The square root of the quotient a b is equal to the quotient of the square roots of a and b , where b 0.

a b = a b

Given a radical expression, use the quotient rule to simplify it.

  1. Write the radical expression as the quotient of two radical expressions.
  2. Simplify the numerator and denominator.

Using the quotient rule to simplify square roots

Simplify the radical expression.

5 36

5 36 Write as quotient of two radical expressions . 5 6 Simplify denominator .

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Simplify 2 x 2 9 y 4 .

x 2 3 y 2 . We do not need the absolute value signs for y 2 because that term will always be nonnegative.

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Using the quotient rule to simplify an expression with two square roots

Simplify the radical expression.

234 x 11 y 26 x 7 y

234 x 11 y 26 x 7 y Combine numerator and denominator into one radical expression . 9 x 4 Simplify fraction . 3 x 2    Simplify square root .

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Simplify 9 a 5 b 14 3 a 4 b 5 .

b 4 3 a b

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Adding and subtracting square roots

We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of 2 and 3 2 is 4 2 . However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression 18 can be written with a 2 in the radicand, as 3 2 , so 2 + 18 = 2 + 3 2 = 4 2 .

Given a radical expression requiring addition or subtraction of square roots, solve.

  1. Simplify each radical expression.
  2. Add or subtract expressions with equal radicands.

Questions & Answers

how does the planets on our solar system orbit
cheten Reply
how many Messier objects are there in space
satish Reply
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Richard Reply
what are astronomy
Issan Reply
Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.
Rafael
vjuvu
Elgoog
what is big bang theory?
Rosemary
what type of activity astronomer do?
Rosemary
No
Richard
the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen
Roaul
I want to be an astronomer. That's my dream
Astrit
Who named the the whole galaxy?
Shola Reply
solar Univers
GPOWER
what is space
Richard
what is the dark matter
Richard
what are the factors upon which the atmosphere is stratified
Nicholas Reply
is the big bang the sun
Folakemi Reply
no
Sokak
bigbang is the beginning of the universe
Sokak
but thats just a theory
Sokak
nothing will happen, don't worry brother.
Vansh
what does comet means
GANGAIN Reply
these are Rocky substances between mars and jupiter
GANGAIN
Comets are cosmic snowballs of frozen gases , rock and dust that orbit the sun. They are mostly found between the orbits of Venus and Mercury.
Aarya
hllo
John
hi
John
qt rrt
John
r u there
John
hey can anyone guide me abt international astronomy olympiad
sahil
how can we learn right and true ?
Govinda Reply
why the moon is always appear in an elliptical shape
Gatjuol Reply
Because when astroid hit the Earth then a piece of elliptical shape of the earth was separated which is now called moon.
Hemen
what's see level?
lidiya Reply
Did you mean eye sight or sea level
Minal
oh sorry it's sea level
lidiya
according to the theory of astronomers why the moon is always appear in an elliptical orbit?
Gatjuol
hi !!! I am new in astronomy.... I have so many questions in mind .... all of scientists of the word they just give opinion only. but they never think true or false ... i respect all of them... I believes whole universe depending on true ...থিউরি
Govinda
hello
Jackson
hi
Elyana
we're all stars and galaxies a part of sun. how can science prove thx with respect old ancient times picture or books..or anything with respect to present time .but we r a part of that universe
w astronomy and cosmology!
Michele
another theory of universe except big ban
Albash Reply
how was universe born
Asmit Reply
there many theory to born universe but what is the reality of big bang theory to born universe
Asmit
what is the exact value of π?
Nagalakshmi
by big bang
universal
there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.
Aarya
I think after Big Bang!
Michele
from where on earth could u observe all the stars during the during the course of an year
Karuna Reply
I think it couldn't possible on earth
Nagalakshmi
in this time i don't Know
Michele
is that so. the question was in the end of this chapter
Karuna
in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*). So it would just barely creep over the horizon.
Christopher
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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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