Multiplying Exponents with Square Roots

In this tutorial, we will learn the multiplication of exponents, where bases have a square root.

When multiplying square roots that contain exponents, we can rewrite the term with a rational exponent.

The square root of a positive number (√a) can be expressed as a rational exponent and

(√a) = (a)¹/².

When we need to rewrite a given exponential term as a rational exponent, we multiply the existing power with 1/2.

Example, if we need to rewrite (√7)³ as a rational exponent, we will first converted radical (√7) to (7)¹/² then we will multiply the power 3 with 1/2 which makes 3/2.

Now the radical (√7)³ is converted to a rational exponent and written as (7)³/².

Example: (√3)² x (√4)⁴

= (3)²/² x (4)⁴/²

= 3 x (4)²

= 3 x 16

= 48

Here notice that new power of base 3 became 1, and new power of base 4 became 2.

Hence our problem became 3 multiplied by 16.

When our exponents do not divide evenly by root 2.

Then we separate our base into two terms, one with a power that divides evenly by two, and one with a power of 1.

In this section, we will explore the multiplication of exponents where the bases have a square root.

The exponents rules remain the same if the bases are square roots.

When the square root bases are same, the powers are added.

Example: Solve (√3)² x (√3)³

Solution: Square root bases are same.

Thus, (√3)² x (√3)³

= (√3)²⁺³

= (√3)⁵

= (3)⁵/²

When the square root bases are different and the powers are same, the bases are multiplied first.

The square root bases are different and the powers are same, first we multiplied the bases.

(√3)³ x (√5)³ =

(√3 x √5)³ = {(√3 x 5)}³ = (√15)³ = (15)³/²

When the square root bases and powers are different, the exponents are evaluated separately and then multiplied.

The square root bases and powers are different.

Thus, (√4)³ x (√3)⁴

= (√4)³ x (√3)⁴

= {(√4) x (√4) x (√4)} x {(√3) x (√3) x (√3) x (√3)}

= (√64) x (√81)

= 8 x 9

= 72