CONVERTING BETWEEN RADICAL FORM AND EXPONENTIAL FORM

Write each expression in radical form.

Problem 1 :

7^1/2

Solution:

= 7^1/2

= √7

Problem 2 :

4^4/3

Solution:

= 4^4/3

Writing 4 in exponential form, we get 4 = 2²

= (2²)^4/3

When we have power raised to another power, we will multiply both the powers.

= 2^{2 × (4/3)}

= 2^8/3

= ∛(2 × 2 × 2 × 2 × 2 × 2 × 2 × 2)

= 4∛4

Problem 3 :

2^5/3

Solution:

= 2^5/3

We can write the fractional power as product of integer and fraction. So, we get

= 2^{5 (1/3)}

1/3 can be written as cube root.

= ∛ (2)⁵

= ∛ (2 × 2 × 2 × 2 × 2)

= 2 ∛4

Problem 4 :

7^4/3

Solution:

= 7^4/3

We can write the fractional power as product of integer and fraction. So, we get

= 7^{4 (1/3)}

1/3 can be written as cube root.

= ∛ (7)⁴

= ∛ (7 × 7 × 7 × 7)

= 7 ∛7

Problem 5 :

6^3/2

Solution:

= 6^3/2

We can write the fractional power as product of integer and fraction. So, we get

= (6)^{3 (1/2)}

1/2 can be written as square root.

= √(6)³

= √(6 × 6 × 6)

= 6√6

Problem 6 :

2^1/6

Solution:

= 2^1/6

Power 1/6 can be written as 6^th root.

= ⁶√2

Write each expression in exponential form.

Problem 7 :

(√10)³

Solution:

Square root can be written as power 1/2.

= (√10³)^1/2

= (10)^3/2

Problem 8 :     

⁶√2 

Solution:

= ⁶√2 

6^th root can be written as power 1/6.

= (2)^1/6

Problem 9 :     

(∜2)⁵

Solution:

= (∜2)⁵

4^th root can be written as power 1/4.

= (2^(1/4))⁵

= (2)^5/4

Problem 10 :     

(∜5)⁵

Solution:

= (∜5)⁵

4^th root can be written as power 1/4.

= (5^(1/4))⁵

= (5)^5/4

Problem 11 :     

∛2

Solution:

= ∛2

cube root can be written as power 1/3.

= (2)^1/3

Problem 12 :     

⁶√10

Solution:

= ⁶√10 

6^th root can be written as power 1/6.

= (10)^1/6

Write each expression in radical form.

Problem 13 :     

(5x)^-5/4

Solution:

Let us write -5/4 as a product of integer and fraction.

(5x)^-5/4 = (5x)^-5 · ^(1/4)

Changing power 1/4 as 4^th root.

= ∜(5x)^-5

Inorder to change the negative exponent as positive exponent, we will flip the base.

= ∜1/(5x)⁵

= 1/∜(5x)⁵

Problem 14 :     

(5x)^-1/2

Solution:

Let us write -1/2 as a product of integer and fraction.

(5x)^-1/2 = (5x)^-1 · ^(1/2)

Changing power 1/2 as square root.

= √(5x)^-1

In order to change the negative exponent as positive exponent, we will flip the base.

= 1/√(5x)

Problem 15 :

(10n)^3/2

Solution:

= (10n)^3/2

Writing the fractional power as a product of integer and fraction, we get

= (10n)^{3 (1/2)}

Power 1/2 can be written as square root.

= √(10n)³

Here 10n is repeated 3 times inside the square root.

= √(10n) (10n) (10n)

= 10n√(10n)

Problem 16 :

a^6/5

Solution:

Writing the fractional power as a product of integer and fraction, we get

= a^6/5

= a^{6 x (1/5)}

1/5 can be written as 5th root.

= ⁵√a⁶