SIMPLIFYING EXPONENTIAL EXPRESSIONS

Finding Product of Powers

Example 1 :

Simplify : 

a⁴ ⋅ b⁵ ⋅ a²

Solution :

=  a⁴ ⋅ b⁵ ⋅ a²

Group powers with the same base together. 

=  (a⁴ ⋅ a²) ⋅ b⁵

Add the exponents of powers with the same base.   

=  a^{4 + 2} ⋅ b⁵

=  a⁶ ⋅ b⁵

Example 2 :

Simplify : 

x² ⋅ x ⋅ x^-4

Solution :

=  x² ⋅ x ⋅ x^-4

Because the powers have the same base, keep the base and add the exponents.  

=  x^{2 + 1 + (-4)}

=  x^{2 + 1 - 4}

=  x^-1

=  1/x

Finding Powers of Powers

Example 3 :

Simplify : 

(y²)^-4 ⋅ y⁵

Solution :

=  (y²)^-4 ⋅ y⁵

Use the Power of a Power Property. 

=  y^-8 ⋅ y⁵

=  y^{-8 + 5}

=  y^-3

=  1/y³

Example 4 :

Simplify : 

(xy²)^-4 ⋅ (x³y⁵)²

Solution :

=  (xy²)^-4 ⋅ (x³y⁵)²

Use the Power of a Power Property. 

=  x^-4(y²)^-4 ⋅ (x³)²(y⁵)²

=  x^-4y^-8 ⋅ x⁶y¹⁰

Group powers with the same base together. 

=  (x^-4 ⋅ x⁶) ⋅ (y^-8 ⋅ y¹⁰)

=  x^{-4 + 6} ⋅ y^{-8 + 10}

=  x²y²

Finding Powers of Product

Example 5 :

Simplify : 

(-5x)²

Solution :

=  (-5x)²

Use the Power of a Product Property. 

=  (-5)² ⋅ x²

=  25x²

Example 6 :

Simplify : 

-(5x)²

Solution :

=  -(5x)²

Use the Power of a Product Property. 

=  -(5² ⋅ x²)

=  -(25 ⋅ x²)

=  -25x²

Example 7 :

Simplify : 

(a^-2 ⋅ b⁰)³

Solution :

=  (a^-2 ⋅ b⁰)³

Use the Power of a Product Property. 

=  (a^-2)³ ⋅ (b⁰)³

=  a^-2 ⋅ 3 ⋅ b^0 ⋅ 3

=  a^-6 ⋅ b⁰

=  x^-6 ⋅ 1

=  1/a⁶

Finding Quotient of Powers

Example 8 :

Simplify : 

xy³/y⁵

Solution :

=  xy³/y⁵

Use the Quotient of Powers Property. 

=  xy^{3 - 5}

=  xy^-2

=  x/y²

Example 9 :

Simplify : 

x⁵y⁹/(xy)⁴

Solution :

=  x⁵y⁹/(xy)⁴

Use the Power of a Product Property. 

=  x⁵y⁹/x⁴y⁴

Use the Quotient of Powers Property. 

=  x^5-4 ⋅ y^9-4

=  x¹ ⋅ y⁵

=  xy⁵

Finding Positive Powers of Quotients

Example 10 :

Simplify : 

(2a³/bc)³

Solution :

=  (2a³/bc)³

Use the Power of a Quotient Property. 

=  (2a³)³/(bc)³

Use the Power of a Power Property. 

=  2³(a³)³/(b³c³)

=  8a⁹/b³c³

Finding Negative Powers of Quotients

Example 11 :

Simplify : 

(3a/b²)^-3

Solution :

=  (3a/b²)^-3

Rewrite with a positive exponent. 

=  (b²/3a)³

Use the Power of a Quotient Property. 

=  (b²)³/(3a)³

Use the Power of a Power Property. 

=  b⁶/(3³a³)

=  b⁶/27a³

Example 12 :

Simplify : 

(3/4)^-1 ⋅ (2a/3b)^-2

Solution :

=  (3/4)^-1 ⋅ (2a/3b)^-2

Rewrite each fraction with a positive exponent. 

=  (4/3)¹ ⋅ (3b/2a)²

Use the Power of a Quotient Property. 

=  (4/3) ⋅ (3b)²/(2a)²

Use the Power of a Power Property. 

=  (4/3) ⋅ (3²b²/2²a²)

=  (4/3) ⋅ (9b²/4a²)

Simplify. 

=  3b²/a²

Example 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 :

2^k ⋅ 8^w = 2²⁰

2^k ⋅ (2)^3w = 2²⁰

2^k+3w = 2²⁰

k + 3w = 20

So, the value of k + 3w is 20 and opiton D is correct.

Example 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 :

x² = 25, y² = 4, and (x + 5)(y - 2) = 0

x = -5 and 5, y = -2 and 2

(x + 5)(y - 2) ≠ 0

• If x = -5 then y = 2 or -2, then (x + 5)(y - 2) = 0
• If y = 2, then x = -5 or x = 5, then (x + 5)(y - 2) = 0

Then the above combinations are not possible.

if x = 5 and y = -2

x³ - y³ = 5³ - (-2)³

= 125 + 8

= 133

So, option E is correct.

Example 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 :

Given that,

= 3^{(a - b)} 3^{(a + b)}/ 3^{(2a + 1)}

Using the rules of exponents,

= 3^{(a - b + a + b)}/ 3^{(2a + 1)}

= 3^2a/ 3^{(2a + 1)}

= 3^{2a - 2a - 1}

= 3^{- 1}

By converting the negative exponent as positive exponent, we get

= 1/3

So, option A is correct.

Example 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 :

Given that,

= 2^{(a - 1) (a + 1)}/ 2^{(a - 2) (a + 2)}

= 2^(a² - 1²) / 2^(a² - 2²)

= 2^(a² - 1² - (a² - 2²))

= 2^(a² - 1² - a² + 2²)

= 2^(- 1 + 4)

= 2³

= 8

So, option C is correct.

Example 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 :

Given that,

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

x^{2 x 1/2} y^{3 x 1/2} x^{2 x 1/3} y^{3 x 1/3} = (x^a/3 y^a/2)

x¹ y^3/2 x^2/3  y¹  = (x^a/3 y^a/2)

x^{1 + (2/3)}  y^{(3/2) + 1}  = (x^a/3 y^a/2)

x^5/3  y^5/2  = (x^a/3 y^a/2)

a/3 = 5/3

Then a = 5, option C is correct.

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