PRACTICE QUESTIONS ON SIMPLIFYING EXPONENTIAL EXPRESSIONS

Express the following in simplest exponential form with a prime number base

1)  4

2)  16

3)  27

4)  4²

5)  25²

6)  2^t × 8

7)  3^a ÷ 3

8)  2ⁿ × 4ⁿ

9)  9/(3^x)

10)  5ⁿ⁺²/5^n–2

11)  (3⁴)^a+1

12)  3^x × 3^4-x

13)  (4^a)/2^b

14)  8^x/16^y

15)  5^1+x /5^x-1

16)  (3^a × 9^a)/27^a+2

Problem 17 :

The following numbers can be written in the form 2ⁿ. Find n.

a)  32      b)  256     c)  4056

Solution

Problem 18 :

By considering 3¹, 3², 3³, 3⁴, ............... and looking for a pattern, find the last digit of 3³³

Solution

Problem 19 :

What is the last digit of 7⁷⁷

Solution

Problem 20 :

Find n, if 

a) 5⁴  = n      b)  n³ = 343      c)  11ⁿ = 161051

Solution

Simplify the following expressions

Questions :

1)  x⁴/x

2)  4b² × 2b³

3)  a⁶b³/ a⁴b

4)  18x⁶/3x³

5)  5x³y²/15xy

6)  24t⁶ r⁴/15t⁶r²

7)  3pq³ × 5p⁵

8)  x¹²/(x³)²

9)  (t⁶ × t⁴)/(t²)³

Solution

Problem 10 :

If (-a² b³) (2ab²) (-3b) = ka^m bⁿ, what is the value of m + n ?

Solution

Problem 11 :

If (2/3 a²b)² (4/3 ab)^-3 = ka^m bⁿ, what is the value of k ?

Solution

Problem 12 :

If [x³ (-y)² z^-2]/x^-2y³z = x^m/yⁿz^p, what is the value of m + n + p?

Solution

Problem 13 :

If 2^x = 5, what is the valu of 2^x + 2^2x + 2^3x ?

Solution

Problem 14 :

(6xy²)(2xy)² / (8x²y²)

If the expression above is written in the form ax^myⁿ, what is the value of a + m + n?

Solution

Problem 15 :

If 8,200 × 300,000  is equal to 2.46 x 10ⁿ, what is the value of n?

Solution

Problem 16 :

(3^x + 3^x + 3^x) 3^x Which of the following is equivalent to the expression shown above?

Solution

Problem 17 :

a + b = 4 and a - b = 2, if x > 1 and x^a²/x^b² = x^c, what is the value of c ?

Solution

Simplify : 

1)  a⁴ ⋅ b⁵ ⋅ a^{2          Solution}

2)  x² ⋅ x ⋅ x^{-4        Solution}

3)  (y²)^-4 ⋅ y^{5          Solution}

4)  (xy²)^-4 ⋅ (x³y⁵)^{2           Solution}

5)  (-5x)^{2          Solution}

6)  -(5x)^{2         Solution}

7)  (a^-2 ⋅ b⁰)^{3     Solution}

8)  xy³/y^{5      Solution}

9)  x⁵y⁹/(xy)^{4        Solution}

10)  (2a³/bc)^{3   Solution}

11)  (3a/b²)^{-3           Solution}

12)  (3/4)^-1 ⋅ (2a/3b)^{-2               Solution}

13)  If 2^k ⋅ 8^w = 2²⁰, what is the value of k + 3w?

(A) 8    (B) 12   (C) 16   (D) 20    (E) 24

Solution

14)  If x² = 25, y² = 4, and (x + 5)(y - 2) ≠ 0, then x³ - y³ =

(A) -133    (B) -117    (C) 117     (D) 125    (E) 133

Solution

15)  What is the value of 3^{(a - b)} 3^{(a + b)}/ 3^{(2a + 1)} ?

(A) 1/3    (B) 1/9    (C) 3     (D) 9

Solution

16)  What is the value of 2^{(a - 1) (a + 1)}/ 2^{(a - 2) (a + 2)} ?

(A) 1/16    (B) 1/8    (C) 8     (D) 16

Solution

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

If the equation above , where a is a constant , is true for aal positive lvaues of x an dy , what is the value of a ?

(A) 2   (B) 3    (C) 5    (D) 6

Solution

Simplify the following expressions involving exponential function

1)  (4^m + 8^m)/4^{m        Solution}

2)  (4^m + 8^m)/(1 + 2^m)          Solution

3)  (7^b + 21^b)/7^{b        Solution}

4)  (4ⁿ⁺² – 4ⁿ)/4^{n       Solution}

5)  (4ⁿ⁺² – 4ⁿ)/15    Solution

6)  (2^m+n – 2ⁿ)/2^{n   Solution}

7)  (3ⁿ⁺² – 3ⁿ)/3^{n+1    Solution}

8)  If ((2/3)a² b)² (4/3)(ab)^-3 = ka^m bⁿ, what is the value of k ?

Solution

9)  If [(x³) (-y)² z^-2] / (x)^-2 (y)³ z = x^m /yⁿ z^p what is the value of m + n + p ?

Solution

10)  If 2^x = 5, what is the value of 2^x + 2^2x + 2^3x ?

Solution

11)  

(3^x + 3^x + 3^x) 3^x

Which of the following is equivalent to the expressions shown above ?

a)  3^4x      b) 3^{3x^2}     c) 3^{1 + 3x}       d) 3^{1 + 2x}

Solution

12)  

(6xy²)(2xy)² / 8x²y²

If the expression above is written in the form ax^m yⁿ, what is the value of m + n ?

Solution

13)  

What is the value of 3^{(a - b)} 3^{(a + b)} / 3^{2a + 1}  ?

a)  1/3    b)  1/9     c)  3    d)  9

Solution

14)  

What is the value of 2^{(a - 1)(a + 1)}/ 2^{(a - 2)(a + 2)}  ?

a)  1/16    b)  1/8     c)  8    d)  16

Solution

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