<< Chapter < Page Chapter >> Page >

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 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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!

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Simplify 50 x 2 x assuming x > 0.

10 | x |

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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

b 4 3 a b

Got questions? Get instant answers now!

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

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
Practice Key Terms 6

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask