EXPONENTS AND POWERS PRACTICE QUESTIONS

Problem 1 :

Find x so that

(-5)^x+1 ⋅ (-5)⁵  =  (-5)⁷

Problem 2 :

(5/3)^-2 ⋅ (5/3)^-14  =  (5/3)^8x

Problem 3 :

Find x so that

(-2)³ ⋅ (-2)^-6  =  (-2)^2x-1

Problem 4 :

Simplify

3⁰+3¹-3^-1

Problem 5 :

Simplify

(-125)^1/3

Problem 6 :

Simplify

(32)^2/5

Problem 7 :

Simplify

(64)^5/6

Problem 8 :

Simplify

(81)^-3/4

Problem 9 :

Simplify

3pq^-1

Problem 10 :

Simplify

(3^x ⋅ 9^x)/(27)^x+2 

Problem 11 :

Simplify

2^x ⋅ 8

Problem 12 :

Simplify

5ⁿ⁺²/5^n-2

Problem 13 :

Simplify

5^1+x/5^x-1

Problem 14 :

Simplify

8^x/16^y

Problem 15 :

Simplify

(5t^-2)^-1

Problem 16 :

Simplify

(xy)³/y^-2

Problem 17 :

Simplify

3^x-2  =  1/9

Problem 17 :

Simplify

3^x-2  =  1/9

Detailed Solution

Problem 1 :

Find x so that

(-5)^x+1 ⋅ (-5)⁵  =  (-5)⁷

Solution :

(-5)^x+1 ⋅ (-5)⁵  =  (-5)⁷

Using the rule of exponent a^m ⋅ aⁿ = a^m+n

(-5) ^x+1+5  =  (-5)⁷

When bases are equal on both sides, we can equate the powers.

(-5) ^x+6  =  (-5)⁷

x+6  =  7

x  =  7 - 6

x  =  1

So, the value of x is 1.

Problem 2 :

(5/3)^-2 ⋅ (5/3)^-14  =  (5/3)^8x

Solution :

(5/3)^-2 ⋅ (5/3)^-14  =  (5/3)^8x

Using the rule of exponent a^m ⋅ aⁿ = a^m+n

(5/3)^-2-14  =  (5/3)^8x

(5/3)^-16  =  (5/3)^8x

When bases are equal on both sides, we can equate the powers.

-16  =  8x

x  =  -16/8

x  =  -2

So, the value of x is -2.

Problem 3 :

Find x so that

(-2)³ ⋅ (-2)^-6  =  (-2)^2x-1

Solution :

(-2)³ ⋅ (-2)^-6  =  (-2)^2x-1

Using the rule of exponent a^m ⋅ aⁿ = a^m+n

(-2)^3-6  =  (-2)^2x-1

(-2)^-3  =  (-2)^2x-1

By equating the powers, we can solve for the variable.

2x-1  =  -3

2x  =  -3+1

2x  =  -2

x  =  -1

So, the value of x is -1.

Problem 4 :

Simplify

3⁰+3¹-3^-1

Solution :

3⁰+3¹-3^-1

Anything to the power 0 is 1. To convert the negative exponent as positive, we have to convert it as it

=  1 + 3 - (1/3)

=  4 - (1/3)

=  (12-1)/3

=  11/3

The answer is 11/3.

Problem 5 :

Simplify

(-125)^1/3

Solution :

(-125)^1/3

Writing 125 is exponential form, we get

-125  =  -5⋅(-5)⋅(-5)  ==>  (-5)³

=  [(-5)³]^1/3

Using the property (a^m)ⁿ = a^mn. When we have power raised by another power, we have to multiply the powers.

=  -5

The answer is -5.

Problem 6 :

Simplify

(32)^2/5

Solution :

(32)^2/5

Writing 32 in exponential form, we get

32  =  2⋅2⋅2⋅2⋅2 ==>  2⁵

=  (2⁵)^2/5

If we have power raised by another power, we will multiply these two powers. Then 5 x 25 will give 2.

=  2²

=  4

The answer is 4.

Problem 7 :

Simplify

(64)^5/6

Solution :

(64)^5/6

Writing 64 in exponential form, we get

64  =  2⋅2⋅2⋅2⋅2⋅2 ==>  2⁶

=  (2⁶)^5/6

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

=  2⁵

=  32

The answer is 32.

Problem 8 :

Simplify

(81)^-3/4

Solution :

(81)^-3/4

Writing 81 in exponential form, we get

81  =  3⋅3⋅3⋅3 ==>  3⁴

=  (3⁴)^-3/4

Power raised by another power here. So, we have to multiply the powers.

=  3^-3

By writing the reciprocal of 3^-3, we can convert the negative exponent as positive.

=  1/3³

=  1/27

The answer is 1/27.

Problem 9 :

Simplify

3pq^-1

Solution :

The variable q is having negative exponent, by changing the negative exponent as positive exponent, we get

3pq^-1  =  3p/q

The answer is 3p/q.

Problem 10 :

Simplify

(3^x ⋅ 9^x)/(27)^x+2 

Solution :

(3^x ⋅ 9^x)/(27)^x+2

We write 9 in exponential form, then 9 = 3²

=  (3^x ⋅ 3^2x)/(3³)^x+2 

=  (3^x+2x)/3^3(x+2) 

=  3^x+2x/3^3x+6 

=  3^(3x-3x-6)

=  3^-6

=  1/3⁶

The answer is 1/3⁶.

Problem 11 :

Simplify

2^x ⋅ 8

Solution :

2^x ⋅ 8

=  2^x ⋅ 2³

=  2^(x+3)

The answer is 2^(x+3).

Problem 12 :

Simplify

5ⁿ⁺²/5^n-2

Solution :

=  5ⁿ⁺²/5^n-2

=  5^(n+2)-(n-2)

=  5^n+2-n+2

=  5⁴

The answer is 5⁴ or 625.

Problem 13 :

Simplify

5^1+x/5^x-1

Solution :

=  5^1+x/5^x-1

=  5^(1+x)-(1+x)

=  5^1+x-1-x

=  5²

So, the answer is 25.

Problem 14 :

Simplify

8^x/16^y

Solution :

=  8^x/16^y

8  =  2³ and 16  =  2⁴

=  2^3x/2^4y

=  2^3x-4y

Problem 15 :

Simplify

(5t^-2)^-1

Solution :

=  (5t^-2)^-1

=  1/5t^-2

=  t²/5

Problem 16 :

Simplify

(xy)³/y^-2

Solution :

=  (xy)³/y^-2

=  x³y³/y^-2

=  x³y³⋅ y²

=  x³y⁵

Problem 17 :

Simplify

3^x-2  =  1/9

Solution :

3^x-2  =  1/3²

3^x-2  =  1⋅3^-2

3^x-2  =  3^-2

x-2  =  -2

x  =  -2+2

x  =  0

Problem 18 :

If (25)¹⁵⁰ = (25x)⁵⁰, then the value of x will be

a) 5³    b)  5⁴     c)  5²     d)  5

Solution :

(25)¹⁵⁰ = (25x)⁵⁰

Since two terms are multiplied, distributing the power. We get

(25)¹⁵⁰ = (25)⁵⁰ x⁵⁰

(25)¹⁵⁰ / (25)⁵⁰ = x⁵⁰

25^150-50 = x⁵⁰

25¹⁰⁰ = x⁵⁰

(25²)⁵⁰ = x⁵⁰

x = 25²

Writing 25 in exponential form, we get

x = (5²)²

x = 5⁴

So, option b is the answer.

Problem 19 :

Find the value of a from the following :

(√9)^-5 x (√3)^-7 = (√3)^-a

a)  11   b)  13   c)  15    d)  17

Solution :

(√9)^-5 x (√3)^-7 = (√3)^-a

(√(3)²)^-5 x (√3)^-7 = (√3)^-a

(3)^-5 x (3)^-7/2 = (3)^-a/2

(3)^-5-(7/2) = (3)^-a/2

By equating the power, we get

-5 - (7/2) = -a/2

-17/2 = -a/2

a = 17

So, the value of a is 17.

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