SOLVING EQAUTIONS WITH VARAIBLE EXPONENTS WORKSHEET

Solve the following equation.

Solution

(13)  

Consider the equation 9^m/9ⁿ = 9²

a. Find two numbers m and n that satisfy the equation.

b. Describe the number of solutions that satisfy the equation. Explain your reasoning.

Solution

(14)  Find the value of x that makes 8^3x/8^{2x + 1} = 8⁹ true. Explain how you four your answer.            Solution

(15)  

Which of the following is equivalent to

7^x ⋅ x⁷ / 7⁷ ⋅ x^x  ?

a)  1      b)  (x - 7)^7/x     c)  (x/7)^{x - 7}   d)  (7/x)^{x - 7}

Solution

(16)  If (3ab²) ⋅ (2a²  b)³  / 8a² b²  = 3a^mbⁿ ? what is the value of m + n ?

Solution

Solving Equations with Variables in Exponents

(1) If 3^a+1 = 3^-a+7, what is the value of a ?     Solution

(2)  If 3^x+2 = y, then what is the value of 3^x in terms of y ?

(a) y+9  (b) y-9  (c) y/3  (d)  y/9

Solution

(3)  If 2a-b = 4, what is the value of 4^a/2^b         Solution

(4)  If 2^x+3 - 2^x = k(2^x), what is the value of k ?          Solution

(5)  If 

√(x√x) = x^a

then what is the value of a ?           Solution

(6) If n³ = x and n⁴ = 20x, where n > 0, what is the value of x ?

Solution

(7) Solve for x, 2(4^2x+1) = 128         Solution

(8)  12^2x-1 = (∜12)^x             Solution

(9) 3^4x+3 = 81^x               Solution

(10) 12 ⋅ 2^x-7 = 24         Solution

(11)  If (-a² b³) ⋅ (2a b²) (-3b) = ka^m bⁿ What is the value of m + n ?

(12)  If (2/3 a² b)² ⋅ (4/3 ab)^-3 = ka^m bⁿ What is the value of k ?

(13)  If (x³) (-y)² z^-2 /(x)^-2 y³ z = x^m / yⁿ z^p What is the value of m + n + p ?

(14)  If 3^(a + b)² / 3^(a - b)² = 243, what is the value of ab ?

a)  5/4   b)  3/2     c)  7/4     d)  2

Answers

(11)  the value of m + n is 9.

(12)   the value of k is 1/12.

(13)  the value of m + n + p is 9.

(14)  ab = 5/4, option a

Solve each equations

Problem 1 :

16^-3r ⋅ 1/4 = 16

Solution

Problem 2 :

64^2x ⋅ 16^-3x = 1/16

Solution

Problem 3 :

36^3x ⋅ (1/36)^2x = 6³

Solution

Problem 4 :

243^-3r = 9^-2r

Solution

Problem 5 :

(1/16)^-3n ⋅ 64^-2n = 1/16

Solution

Problem 6 :

4^-p ⋅ 4^3p = 4³

Solution

Problem 7 :

25^-x ⋅ 625 = 25

Solution

Problem 8 :

36^3r = 216

Solution

Problem 9 :

8^2m ⋅ 32^m = 16

Solution

Problem 10 :

6^{2n + 2} = 36

Solution

Problem 11 :

9^-v = 81

Solution

Problem 12 :

6³ⁿ = 6³ⁿ

Solution

Problem 13 :

A bacteria culture triples in size every hour. You begin observing the number of bacteria 2 hours after the culture is prepared. The amount y of bacteria x hours after the culture is prepared is represented by y = 162 (3^{x - 2}). When will there be 8100 bacteria ?

Solution

Problem 1 :

If a and b are positive integers and 5^{a + b} = 5^3c, which of the following is equal to 5^9c ?

a)  5^{3a + b}      b) 15^{a + b}        c) 25^{3a + 3b}        d)  125^{a + b}     e) 125^{3a + 3b}

Solution

Problem 2 :

(x² y³)^1/2 (x² y³)^1/3 = x^a/3  y^a/2

If the equation above, where a is constant is true for all positive values of x and y, what is the value of a ?

a)  2      b)  3      c)  5      d)  6

Solution

Problem 3 :

If 49 = (7y)² then y could equal which of the following 

I   -1       II     1     III   7

a)  II only     b)  III only      c)  I and II      d)  I and III      d)  II and III

Solution

Problem 4 :

If (a⁶ a⁶)^k = (a³)^{k + 5} , where a is a constant greater than 1, what is the value of k ?

a)  3/2      b)  5/3     c)  2       d)  3     e)  5

Solution

Problem 5 :

If t and w are positive numbers and t⁸ / t² = w¹², then w⁸ / w²

a)  t²      b)  t³       c)  t⁴        d)  t⁶       e)  t¹²

Solution

Problem 6 :

If t > 0 and t² − 4 = 0 , what is the value of t ?

Solution

Problem 7 :

If a²  = 4, then a⁶ = 

a)  12      b)  16       c)  24        d)  32       e)  64

Solution

Problem 8 :

k^ (x² + xy) k^ (y² + xy) = k²⁵

In the equation above, k > 1 and x = 3 what is the positive value of y ?

a)  1      b)  2       c)  4        d)  5

Solution

Problem 9 :

If 3x - y = 12, what is the value of 8^x/2^y ?

a)  2¹²      b)  44       c)  82 

d) The value cannot be determined from the information given

Solution

Problem 10 :

25^x/5^x+3 = (1/125)^{x + 1}

Solution