EVALUATING EXPRESSIONS WITH NEGATIVE EXPONENTS PRACTICE

Simplify the following expressions

1)  5^{-2             Solution}

2)  25^{-1/2    Solution}

3)  (16/25)^{-3/2      Solution}

4)  (4y)^{-2     Solution}

5)  (27b)^1/3 / (9b)^{-1/2     Solution}

6)  b^-1/2 = 4, what is the value of b ?    Solution

7)  (2^m)^-6 = 16, what is the value of 2³ x m ?    Solution

8)  If 4ⁿ = 20, then what is the value of 4^-n ?     Solution        

9)  If a^-1/2 = 3, what is the value of a ?    Solution

10)  If 3^x = 10, what is the value of 3^x-3 ?    Solution

11)  A drop of water leaks from a faucet every second. How many liters of water leak from the faucet in 1 hour?

Solution

12)

a)  If a is a nonzero number, does the value of a⁰ depend on the value of a?

b)  Explain how to evaluate 10⁻³

c)  Without evaluating, order 5⁰, 5⁴, and 5⁻⁵ from least to greatest.

Solution

13)

Which is different? Find “both” answers.

a)  Rewrite 1/(3 ⋅ 3 ⋅ 3)   using a negative exponent.

b)  Write 3 to the negative third power.

c)  Write 1/3 cubed as a power.

d)  Write (−3) ⋅ (−3) ⋅ (−3) as a power.

Solution

14)  

Use the Quotient of Powers Property to copy and complete the table.

a) What patterns do you see?

b)  How would you define 5⁰? Why?

c) How can you rewrite 5⁻¹ as a fraction?

Solution

15)  Describe and correct the error in evaluating the expression.

Solution

Evaluate the following expressions

1)  yx^1/3 ⋅ xy^{3/2    Solution}

2)  4v^2/3 ⋅ v^{-1    Solution}

3)  (a^1/2 b^1/2)^{-1    Solution}

4)  (x⁰y^1/3)^3/2 x^{0    Solution}

5)  (2x^1/2y^1/3)/(2x^4/3y^-7/4)    Solution

6)  u^-5/4v² ⋅ (u^3/2)^{-3/2    Solution}

7)  uv ⋅ u ⋅ (v^3/2)^{3       Solution}

8)  (2x^-2y^5/3)/x^-5/4y^-5/3 ⋅ xy^{1/2    Solution}

9)  (a^3/4b^-1 ⋅ b^7/4)/3b^{-1      Solution}

10)  (3y^-5/4)/y^-1 ⋅ 2y^{-1/3            Solution}

11)  If x^3/5 = m and y^-3/5 = n^-1, which of the following is equal to xy ?

a)  (mn)^3/5      b)  (mn)^9/25        c)  m^5/3 n^-5/3    d)  (mn)^5/3       e) n/m

 Solution

12)

x⁴ x⁶/x⁸ = 81

According to the equation above, which of the following could equal to x² ?

a)  81     b)  72      c)  18    d)  9     e)  3

 Solution

13)

If x⁶  = 60 and w¹⁰ =  20, what is x¹² w^-10 ?

a)  36     b)  60      c)  120    d)  180     e)  360

 Solution

14)

If a² = b, what is a⁴ in terms of b ?

a)  2b     b)  b + 2      c)  2b + 2    d)  b²     e)  4b

 Solution

15)

If x³ = y⁹, what is x in terms of y ?

a)  √y     b)  y²      c)  y³     d)  y⁶      e)  y¹² 

 Solution

Simplify. Your answer should contain only positive exponents.

Problem 1 :

(x^-2 x^-3)⁴

Solution

Problem 2 :

(x⁴)^-3 ⋅ 2x⁴

Solution

Problem 3 :

(n³)³ ⋅ 2n^-1

Solution

Problem 4 :

(2v)²  ⋅ 2v²

Solution

Problem 5 :

(2x² y  · 4x² y⁴ · 3x)/3x^-3 y²

Solution

Problem 6 :

(2 y³· 3 x y³) / 3 x² y⁴

Solution

Problem 7 :

(x³· y³· x³) / 4 x²

Solution

Problem 8 :

(3 x²· y²) / (2 x^-1 · 4 yx²)

Solution

Problem 9 :

x / (2 x⁰)²

Solution

Problem 10 :

2m^-4 / (2m^-4)³

Solution

Problem 11 :

(2m²)^-1 / m²

Solution

Problem 12 :

(2x³)/(x^-1)³

Solution

Problem 13 :

(a^-3 b^-3)⁰

Solution

Problem 14 :

x⁴ y³ ⋅ (2y²)⁰

Solution

Problem 15 :

ba⁴ ⋅ (2ba⁴)^-3

Solution

Problem 16 :

(2x⁰y²)^-3 ⋅ 2yx³

Solution

Problem 17 :

Express (15³)^-16 as single exponent of 15.

Solution

Problem 18 :

By what number should (7^-2) be multiplied so that the product may be equal to (343)^-1 ?

Solution

Problem 19 :

By what number should (-3/4)⁵ be multiplied so that the product may be equal to (-64/27)^-1 ?     Solution

Problem 20 :

By what number should (-512)^-1 be divided so that the quotient may be equal to 

8^-2 ?

Solution

Problem 21 :

By what number should (144/225)^-1 be divided so that the quotient may be equal to 

(12/15)^-4 ?

Solution

Problem 22 :

Simplify 6^-2 + (3/2)^-2

Solution

Problem 23 :

Find the value of x if [(3/7)³]^-2 = (3/7)^2x

Solution

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

Problem 1 : 

(3x²)³/x¹⁵

(a) 27/x²¹  (b)  27/x¹⁰  (c)  3/x⁹  (d)  27/x⁹

Solution

Problem 2 : 

(-a)^-18

(a) 1/a¹⁸  (b)  18a  (c)  1/-a¹⁸  (d)  1/a^-18

Solution

Problem 3 : 

x^-16/x^-4

(a) 1/x¹²  (b) x¹²  (c)  1/x²⁰  (d)  -x²⁰

Solution

Problem 4 : 

(-3w³/x)⁴

(a) -81w¹²/x⁴          (b) -81w¹²/x        (c)  81w¹²/x⁴          (d)  81w⁷/x⁴

Solution

Problem 5 : 

m^-9 m⁵ m^-1

(a) 1/m⁵    (b) 1/m⁴   (c)  m⁷   (d)  m⁵

Solution

Problem 6 : 

(2^-2 ⋅ 5^-5)^-4

(a) 2⁸ ⋅ 5²⁰    (b) 1/(2⁷ ⋅ 5⁷)  (c)  1/(2⁸ ⋅ 5²⁰)   (d)  2⁷ ⋅ 5⁷

Solution

Problem 7 : 

By what number should (–3)^{– 2} be multiplied so that the product may be equal to 9?

Solution

Problem 8 : 

Find the value of x so that

(5/3)^-2 ⋅ (5/3)^-14 = (5/3)^8x

Solution

Problem 9 : 

If 2^{2x – 3} = (64)^x, find the value of x.

Solution

Problem 10 : 

Find the value of x so that

(-7/11)^-3 ⋅ (-7/11)^5x = [(-7/11)^-2]^-1

Solution

Problem 11 : 

Find the value of x so that

(3/7)^{-2x + 1} / (3/7)^-1 = [(3/7)^-1]^-7

Solution

Problem 12 : 

If p/q = (5/2)^-2 x (4/3)⁰ , find the value of (p/q)^-2

Solution