MULTIPLICATION PROPERTIES OF EXPONENTS WORKSHEET

Simplify :

1)  3² ⋅ 3⁵

2)  5² ⋅ 4^-2 ⋅ 5⁵ ⋅ 4⁶

3)  a⁴ ⋅ b⁵ ⋅ a²

4)  x² ⋅ x ⋅ x^-4

5)  (8⁴)⁵

6)  (5⁶)⁰

7)  (y²)^-4 ⋅ y⁵

8)  (-5x)²

9)  -(5x)²

10)  (a^-2 ⋅ b⁰)³

11)  (p²q³)⁴ ⋅ (p²q⁴)^-4

12)  Light from the Sun travels at about 1.86 x 10⁵ miles per second. It takes about 500 seconds for the light to reach  Earth. Find the approximate distance from the Sun to Earth. Give your answer in scientific notation.  

Detailed Answer Key

Answer (1) :

=  3² ⋅ 3⁵

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

=  3^{2 + 3}

=  3⁵

Answer (2) :

=  5² ⋅ 4^-2 ⋅ 5⁵ ⋅ 4⁶

Group powers with the same base together. 

=  (5² ⋅ 5⁵) ⋅ (4^-2 ⋅ 4⁶)

Add the exponents of powers with the same base.   

=  5^{2 + 5} ⋅ 4^{-2 + 6}

=  5⁷ ⋅ 4⁴

Answer (3) :

=  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

Answer (4) :

=  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

Answer (5) :

=  (8⁴)⁵

Use the Power of a Power Property. 

=  8^4 ⋅ 5

=  8²⁰

Answer (6) :

=  (5⁶)⁰

Use the Power of a Power Property. 

=  5^{6 ⋅ 0}

=  5⁰

=  1

Answer (7) :

=  (y²)^-4 ⋅ y⁵

Use the Power of a Power Property. 

=  y^-8 ⋅ y⁵

=  y^{-8 + 5}

=  y^-3

=  1/y³

Answer (8) :

=  (-5x)²

Use the Power of a Product Property. 

=  (-5)² ⋅ x²

=  25x²

Answer (9) :

=  -(5x)²

Use the Power of a Product Property. 

=  -(5² ⋅ x²)

=  -(25 ⋅ x²)

=  -25x²

Answer (10) :

=  (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

Answer (11) :

=  (p²q³)⁴ ⋅ (p²q⁴)^-4

Use the Power of a Power Property. 

=  (p²)⁴ ⋅ (q³)⁴ ⋅ (p²)^-4 ⋅ (q⁴)^-4

=  p⁸ ⋅ q¹² ⋅ p^-8 ⋅ q^-16

Group powers with the same base together. 

=  (p⁸ ⋅ p^-8) ⋅ (q¹² ⋅ q^-16)

Add the exponents of powers with the same base.   

=  p^{8 + (-8)} ⋅ q^{12 + (-16)}

=  p^{8 - 8} ⋅ q^{12 - 16}

=  p⁰ ⋅ q^-4

=  1 ⋅ q^-4

=  1/q⁴

Answer (12) :

Distance  =  Rate x Time

=  (1.86 x 10⁵) x 500

Write 500 in scientific notation. 

=  (1.86 x 10⁵) x (5 x 10²)

Use the Commutative and Associative Properties to group.  

=  (1.86 x 5) x (10⁵ x 10²)

Multiply within each group.

=  9.3 x 10^{5 + 2}

=  9.3 x 10⁷

The Sun is about 9.3 x 10⁷ miles from Earth. 

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