PROBLEMS INVOLVING RATIONAL EXPONENTS AND RADICALS

Simplify :

Problem 1 :

4^1/2

Solution :

=  4^1/2

By writing 4 in exponential form, we get 4  =  2²

=  (2²)^1/2

=  2 ^{2 x (1/2)}

=  2

So, the answer is 2.

Problem 2 :

225^1/2

Solution :

=  225^1/2

By writing 225 in exponential form, we get 225  =  15²

=  (15²)^1/2

=  15 ^{2 x (1/2)}

=  15

So, the answer is 15.

Problem 3 :

64^1/3

Solution :

=  64^1/3

By writing 64 in exponential form, we get 64  =  4³

=  (4³)^1/3

=  4 ^{3 x (1/3)}

=  4

So, the answer is 4.

Problem 4 :

1000^1/3

Solution :

=  1000^1/3

By writing 1000 in exponential form, we get 1000  =  10³

=  (10³)^1/3

=  10^{3 x (1/3)}

=  10

So, the answer is 10.

Problem 5 :

81^1/4

Solution :

=  81^1/4

By writing 81 in exponential form, we get 81  =  3⁴

=  (3⁴)^1/4

=  3^{4 x (1/4)}

=  3

So, the answer is 3.

Problem 6 :

32^1/5

Solution :

=  32^1/5

By writing 32 in exponential form, we get 32  =  2⁵

=  (2⁵)^1/5

=  2^{5 x (1/5)}

=  2

So, the answer is 2.

Problem 7 :

8^-1/3

Solution :

=  8^-1/3

By writing 8 in exponential form, we get 8  =  2³

=  (2³)^-1/3

=  2^{3 x (-1/3)}

=  2^-1

=  1/2

So, the answer is 1/2.

Problem 8 :

16^3/4

Solution :

=  16^3/4

By writing 64 in exponential form, we get 16  =  2⁴

=  (2⁴)^3/4

=  2^{4 x (3/4)}

=  2³

=  8

So, the answer is 8.

Problem 9 :

27^-2/3

Solution :

=  27^-2/3

By writing 27 in exponential form, we get 27  =  3³

=  (3³)^-2/3

=  3^{3 x (-2/3)}

=  3^-2

=  1/9

So, the answer is 1/9.

Problem 10 :

(2  1/4)^-1/2

Solution :

=  (2  1/4)^-1/2

By converting the mixed fraction into improper fraction, we get

=  (9/4)^-1/2

=  ((3/2)²)^-1/2

=  (3/2)^-1

By flipping the base, we can change the negative exponent as positive.

=  2/3

Problem 11 :

(3  3/8)^-2/3

Solution :

=  (3  3/8)^-2/3

By converting the mixed fraction into improper fraction, we get

=  (27/8)^-2/3

=  ((3/2)³)^-2/3

=  (3/2)^-2

By flipping the base, we can change the negative exponent as positive.

=  (2/3)²

=  4/9

So, the answer is 4/9.

Problem 12 :

Which expression is equivalent to

(x²y)(x⁴y^–3)

where x, y and z are positive numbers ?

A) x⁶y^–3     B) x⁶y^–2     C) x⁸y^–3    D) x⁸y^–2

Solution :

= (x²y)(x⁴y^–3)

= x² x⁴ y y^–3

Since we have same base and they are multiplied, so we have to put the same base and add the exponents

= x²⁺⁴  y^–3+1

= x⁶  y^–2

Problem 13 :

Which expression is equivalent to

√(x/64) for all x > 0

A) x²/8     B) x²/32     C) √x/8    D) √x/32

Solution :

= √(x/64)

Since we have radical for the fraction, we may distribute the radical for the numerator and denominator separately.

= √x/√64

= √x/√(8 ⋅ 8)

= √x/8

So, option C is correct.

Problem 14 :

Which expression is equivalent to

Which expression is equivalent to x^(5/6) / ∛x where x ≠ 0

A) x^1/2     B) x^5/2     C) x^2/3    D) x^5/3

Solution :

= x^(5/6) / ∛x

= x^(5/6) / x^(1/3)

Since we have same bases for both numerator and denominator, we have to put one base and combine the powers.

= x^(5/6) - (1/3)

= x^(5/6) - (2/6)

= x^(5-2)/6

= x^3/6

= x^1/2

Problem 15 :

If (5³)^4k = (5^1/3) ²⁴, what is the value of k ?

Solution :

(5³)^4k = (5^1/3) ²⁴

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

5^3(4k) = 5^{24 (1/3)}

5^12k = 5⁸

By equating the power, we get

12k = 8

k = 8/12

k = 2/3

Problem 16 :

If a = 3√7/4 and 4a = √3b, what is the value of b ?

Solution :

Given that,

a = 3√7/4 -----(1) and 4a = √(3b) ------(2)

4a = 4(3√7/4)

4a = 3√7 

Applying the value of 4a in (2), we get

3√7 =  √(3b)

Squaring both sides

9(7) = 3b

63 = 3b

Dividing by 3 on both sides

b = 63/3

b = 21

So, the value of b is 21.

Problem 17 :

√(2c² - 4) + d = 1

If in the equation above, c > 0 and d = -1, what is the value of c ?

Solution :

√(2c² - 4) + d = 1

Applying the value of d, we get

√(2c² - 4) + (-1) = 1

√(2c² - 4) = 1 + 1

√(2c² - 4) = 2

Squaring on both sides

(2c² - 4) = 4

2c² = 4 + 4

2c² = 8

c² = 4

c = -2 and 2

So, the values of c are -2 and 2.

Problem 18 :

The expression is equivalent to a^2/9(a^2/3)^2/3, where a is positive?

Solution :

= a^2/9(a^2/3)^2/3

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

= a^2/9(a^4/9)

Since we have same bases and they are multiplied, we have to put only one base and combine the powers.

= a^{2/9 + 4/9}

= a^6/9

= a^2/3

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