SIMPLIFY THE FOLLOWING EXPONENTIAL EXPRESSION

Decide whether the expression has been simplified correctly.

Problem 1 :

 (ab)⁸ = ab⁸

A) Yes          B) No

Solution :

Option B) is correct.

Product rule :

(ab)^m = a^m ⋅ b^m

(ab)⁸ = a⁸ ⋅ b⁸

But the given expression is (ab)⁸ = ab⁸ not correct.

The expression has been not simplified correctly.

Problem 2 :

(a/4)⁵ = a⁵/4

A) Yes          B) No

Solution :

Option B) is correct.

(a/4)⁵ = a⁵/4

(a/4)⁵ = a⁵/4⁵

But the given expression is (a/4)⁵ = a⁵/4 not correct.

The expression has been not simplified correctly.

Apply the product rule for exponents, if possible.

Problem 3 :

(-3x⁵y)(-4x⁹y²)

A) 12x⁴⁵y²      B) -12x¹⁴y²      C) 12x¹⁵y³       D)12x¹⁴y³

Solution :

= (-3x⁵y)(-4x⁹y²)

= (-3) ⋅ (-4) ⋅ x⁵ ⋅ x⁹ ⋅ y ⋅ y²

= 12x¹⁴y³

Option D) is correct.

Evaluate the expression.

Problem 4 :

10⁰ + 5⁰

A) 0      B) 1       C) 15      D)2

Solution :

= 10⁰ + 5⁰

= 1 + 1

= 2

Option D) is correct.

Problem 5 :

-10⁰

A) -10      B) 1       C) -1        D)0

Solution :

-10⁰

Any thing to the power of zero is 1.

= -1

Option C) is correct.

Write the expression with only positive exponents. Assume all variables represent nonzero numbers. Simplify if necessary.

Problem 6 :

5x^-2

A) 1/5x²        B) -10x         C) 5/x²        D)1/25x²

Solution :

5x^-2

= 5 × 1/x²

= 5/x²

Option C) is correct.

Problem 7 :

(-a)^-18

A) 1/a¹⁸         B) 18a          C) 1/-a¹⁸       D)1/a^-18

Solution :

(-a)^-18

= a^-18

a^-n = 1/aⁿ

= 1/a¹⁸

Option A) is correct.

Evaluate the expression.

Problem 8 :

5^-4/6^-3

A) 1296/3125         B) 216/625     

C) 3125/1296       D)625/216

Solution :

5^-4/6^-3

a^-n = 1/aⁿ

= 6³/5³

= 216/625

Option B) is correct.

Problem 9 :

1/-3^-3

A) -9      B) -27        C) 9        D)27

Solution :

1/-3^-3

= -1/3^-3

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

= -3³

= -27

Option B) is correct.

Problem 10 :

(2/7)^-3

A) -343/8      B) 8/343      C) -8/343     D)343/8

Solution :

(2/7)^-3

Since we have negative exponents, by taking the reciprocal we can convert it as reciprocal.

= (7/2)³

= 343/8

Option D) is correct.

Problem 11 :

For a non zero integer x, x⁷ ÷ x^{12 is equal to} 

A) x⁵         B) x¹⁹         C) x^-5        D) x^-19 

Solution :

= x⁷ ÷ x¹²

=  x^7-12

= x^-5

By converting the negative exponent as positive exponent, we get

= 1/x⁵

So, option C is correct.

Problem 12 :

The value of (7^–1 – 8^–1)^–1 – (3^–1 – 4^–1)^–1 is

(A) 44      (B) 56       (C) 68        (D) 12

Solution :

= (7^–1 – 8^–1)^–1 – (3^–1 – 4^–1)^–1 

= (1/7 - (1/8))^–1 – (1/3 – 1/4)^–1 

= ((8 - 7)/56)^–1 – ((4 - 3)/12)^–1 

= (1/56)^–1 – (1/12)^–1 

To convert the negative exponent as positive exponent, we convert into its reciprocal.

= 56 - 12

= 44

So, option A is correct.

Problem 13 :

((2/13)^–6  ÷ (2/13)³)³ x (2/13)^–9

Solution :

= ((2/13)^–6  ÷ (2/13)³)³ x (2/13)^–9

= ((13/2)⁶ ÷ (2/13)³)³ x (13/2)⁹

= ((13/2)⁶  ÷ (2/13)³)³ x (13/2)⁹

= (13/2)¹⁸  ÷ (2/13)⁹ x (13/2)⁹

= (13/2)¹⁸  ÷ (13/2)^-9 x (13/2)⁹

= (13/2)^{18 + 9} x (13/2)⁹

= (13/2)²⁷ x (13/2)⁹

= (13/2)²⁷⁺⁹

= (13/2)³⁶

Problem 14 :

The value of [3^-1 × 4^-1]² is _________

Solution :

= [3^-1 × 4^-1]²

= (1/3 x 1/4)²

= (1/(3 x 4))²

= (1/12)²

= 1/144

So, the answer is 1/144.

Problem 15 :

The value of [2^–1 × 3^–1]^–1 is _________.

Solution :

= [2^–1 × 3^–1]^–1

=  [1/2 × 1/3]^–1

=  [1/(2 x 3)]^–1

=  [1/6]^–1

= [6/1]¹

= 6

So, the answer is 6.

Problem 16 :

By solving (6⁰ – 7⁰) × (6⁰ + 7⁰) we get ________.

Solution :

= (6⁰ – 7⁰) × (6⁰ + 7⁰)

Anything to the power of 0 is 1.

= (1 - 1) x (1 + 1)

= 0 x 2

= 0

So, the answer is 0.

Problem 17 :

The expression for 35 with a negative exponent is _________.

Solution :

35 = 7 x 5

= (1/7 x 1/5)^-1

= (1/35)^-1

= (35/1)¹

= 35

So, the required expression is  (1/7 x 1/5)^-1

Problem 18 :

The value for (–7)⁶ ÷ 7⁶ is _________.

Solution :

= (–7)⁶ ÷ 7⁶

= (-7/7)⁶

= (-1)⁶

= 1

So, the answer is 1.

Problem 19 :

The value of [1^–2 + 2^–2 + 3^–2] × 6² is ________ .

Solution :

= [1^–2 + 2^–2 + 3^–2] × 6²

= [1/1² + 1/2² + 1/3²] × 6²

= [1/1 + 1/4 + 1/9] × 36

= [(36 + 9 + 4)/36] x 36

= (49 / 36) x 36

= 49

So, the answer is 49.

Problem 20 :

Simplify the following [49 ⋅ z^–3] / 7^–3 ⋅ 10 ⋅ z^–5] 

Solution :

= [49 ⋅ z^–3] / 7^–3 ⋅ 10 ⋅ z^–5] 

= [49/z³] / (1/7³ x 10 x 1/z⁵] 

= [7²/z³] / (1/7³ x 10 x 1/z⁵] 

= [7² 7³/z³] / [10 x 1/z⁵] 

= [7²⁺³z⁵]/z³ / 10

= [7⁵z^5-3] / [10] 

= 7⁵z² / 10

So, the answer is 7⁵z² / 10.

Problem 21 :

Find the value of x^–3 if x = (100)^1–4 ÷ (100)⁰.

Solution :

First simplifying, 

= (100)^1–4 ÷ (100)⁰

= (100)^-3 ÷ (100)⁰

= 1/(100)³ ÷ 1

= 1/(100)³

= (100)^-3

Comparing with x^–3,

x = 100

so, the value of x is 100.