When multiplying square roots that contain exponents, we must understand how powers and exponents work together. When an exponent is contained within a square root, we can rewrite the term with a rational exponent. For this rational exponent, we will use current exponent as the numerator and the root of 2 for the denominator.

Ex:

Notice that our new power for the base of 3 became 1, and the new power for the base of 4 became 2. Hence, our problem became 3 multiplied by 16.

Sometimes, our exponents do not divide evenly by our root of 2. When this happens, a base with the power of 1 will have to remain in the radical. To work these problems, we separate our base into two terms, one with a power that divides evenly by two, and one with a power of 1.

Example:


Notice a √5 and a √2 had to stay in the radicals because their powers did not divide evenly by our root of two.

Finally, we must be able to this with variables as well.

Example:


We had to break up numbers as well as variables in order to bring out as much as possible. Notice that after simplifying, we combined the two terms by multiplying the coefficients together, as well as our bases.

Practice Problems

Simplify.

1. √6⁴ x √3⁸

2. √7³ x √3⁵

3. 4√(2⁶y³) x 5√(4⁵z⁵)

Answers

1. √6⁴ x √3⁸

6^4/2 x 3^8/2

6² x 3⁴

36 x 81

2916


2. √7³ x √3⁵

√7²7¹ x √3⁴3¹

7^2/2 √7 x 3^4/2√3

7¹ √7 x 3²√3

(7 x 9) √(7 x 3)

63 √21


3. 4√(2⁶y³) x 5√(4⁵z⁵)

4√(2⁶y²y¹) x 5√(4⁴4¹z⁴z¹)

4(2^6/2)(y^2/2)√y x 5(4^4/2)(z^4/2)√(4z)

4(8)(y) √y x 5(16)(z²) √(4z)

32y √y x 80z² √4z

(32 x 80)y z² √( y x 4z)

2,560 yz² √4yz

Related Links: Math algebra Dividing with Square Roots Introduction to Cube Roots Simplifying Square Roots Multiplying Square Roots