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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 look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
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 √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply
Practice Key Terms 6

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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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