Exponential Expressions: Multiplying and Dividing Worksheet | Fun ... - Free Printable

Let’s solve each section step by step.

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Section A: Find the values

1) \( 3^0 \)
Any number to the power of 0 is 1.
→ Answer: 1

2) \( (-2)^3 \)
This means: \( -2 \times -2 \times -2 = 4 \times -2 = -8 \)
→ Answer: -8

3) \( (-1)^{20} \)
Even exponent → positive result.
\( (-1)^{even} = 1 \)
→ Answer: 1

4) \( 7^3 \)
\( 7 \times 7 = 49 \), then \( 49 \times 7 = 343 \)
→ Answer: 343

5) \( (-5)^3 + (-4)^2 \)
First: \( (-5)^3 = -125 \)
Second: \( (-4)^2 = 16 \)
Add: \( -125 + 16 = -109 \)
→ Answer: -109

6) \( \left(\frac{3}{4}\right)^2 \)
Square numerator and denominator: \( \frac{3^2}{4^2} = \frac{9}{16} \)
→ Answer: \( \frac{9}{16} \)

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Section B: Simplify (use exponent rules)

Rule reminder:
- Multiplying same base? Add exponents: \( a^m \times a^n = a^{m+n} \)
- Dividing same base? Subtract exponents: \( a^m ÷ a^n = a^{m-n} \)
- Any number to power 0 is 1.

1) Already done: \( 5^2 × 5^4 = 5^6 \)

2) \( 9^{11} × 9^2 = 9^{11+2} = 9^{13} \)
→ Answer: \( 9^{13} \)

3) \( 2^5 × 2 = 2^5 × 2^1 = 2^{5+1} = 2^6 \)
→ Answer: \( 2^6 \)

4) \( 8^5 × 8^0 = 8^5 × 1 = 8^5 \)
→ Answer: \( 8^5 \)

5) \( 7^5 ÷ 7^3 = 7^{5-3} = 7^2 \)
→ Answer: \( 7^2 \)

6) \( 12^{10} ÷ 12^6 = 12^{10-6} = 12^4 \)
→ Answer: \( 12^4 \)

7) \( 4^{21} ÷ 4 = 4^{21} ÷ 4^1 = 4^{20} \)
→ Answer: \( 4^{20} \)

8) \( 5^3 ÷ 5^3 = 5^{3-3} = 5^0 = 1 \)
→ Answer: 1

9) \( 2^{-3} × 2^8 = 2^{-3+8} = 2^5 \)
→ Answer: \( 2^5 \)

10) \( 6^4 ÷ 6^{12} = 6^{4-12} = 6^{-8} \)
→ Answer: \( 6^{-8} \)

11) \( 15^2 × 15^{-9} = 15^{2 + (-9)} = 15^{-7} \)
→ Answer: \( 15^{-7} \)

12) \( 7^{10} ÷ 7^{14} × 7 \)
First: \( 7^{10} ÷ 7^{14} = 7^{-4} \)
Then: \( 7^{-4} × 7^1 = 7^{-3} \)
→ Answer: \( 7^{-3} \)

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Section C: Simplify fractions with exponents

Use rule: \( \frac{a^m}{a^n} = a^{m-n} \)

1) \( \frac{2^{10}}{2^4} = 2^{10-4} = 2^6 \)
→ Answer: \( 2^6 \)

2) \( \frac{8^{-1}}{8^2} = 8^{-1 - 2} = 8^{-3} \)
→ Answer: \( 8^{-3} \)

3) \( \frac{10^{-12}}{10^7} = 10^{-12 - 7} = 10^{-19} \)
→ Answer: \( 10^{-19} \)

4) \( \frac{3^{-6}}{3^{-1}} = 3^{-6 - (-1)} = 3^{-5} \)
→ Answer: \( 3^{-5} \)

5) \( \frac{3^3 × 3^5}{3^2} = \frac{3^{8}}{3^2} = 3^{6} \)
→ Answer: \( 3^6 \)

6) \( \frac{6^{-2} × 6^4}{6^{-8}} = \frac{6^{2}}{6^{-8}} = 6^{2 - (-8)} = 6^{10} \)
→ Answer: \( 6^{10} \)

7) \( \frac{7^{-8} × 7^{-7}}{7^{-4}} = \frac{7^{-15}}{7^{-4}} = 7^{-15 - (-4)} = 7^{-11} \)
→ Answer: \( 7^{-11} \)

8) \( \frac{14^{-3}}{14^{-11} × 14^{-19}} = \frac{14^{-3}}{14^{-30}} = 14^{-3 - (-30)} = 14^{27} \)
→ Answer: \( 14^{27} \)

9) \( \frac{4^8 × 4^5}{4^2 × 4^4} = \frac{4^{13}}{4^6} = 4^{7} \)
→ Answer: \( 4^7 \)

10) \( \frac{6^{-7} × 6^3}{6^{-1} × 6} = \frac{6^{-4}}{6^{0}} = 6^{-4} \)
Wait — let’s check denominator: \( 6^{-1} × 6^1 = 6^{0} = 1 \)
Numerator: \( 6^{-7+3} = 6^{-4} \)
So overall: \( 6^{-4} ÷ 1 = 6^{-4} \)
→ Answer: \( 6^{-4} \)

11) \( \frac{20^{-17} × 20^9}{20^4 × 20^{-11}} = \frac{20^{-8}}{20^{-7}} = 20^{-8 - (-7)} = 20^{-1} \)
→ Answer: \( 20^{-1} \)

12) \( \frac{5e^8 × 4e^5}{2e^2 × 10e^4} \)
First, multiply coefficients: numerator: \( 5×4 = 20 \), denominator: \( 2×10 = 20 \) → cancels to 1
Exponents on e: numerator: \( e^{8+5} = e^{13} \), denominator: \( e^{2+4} = e^6 \)
So: \( \frac{e^{13}}{e^6} = e^7 \)
→ Answer: \( e^7 \)

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Extension: Investigate and simplify

Pattern: \( (a^m)^n = a^{m×n} \)

Left column:

(2¹)² = 2² → given
(2²)² = 2⁴
(2³)² = 2⁶
(2⁴)² = 2⁸
(2⁵)² = 2¹⁰

Middle column:

(5¹)³ = 5³ → given
(5²)³ = 5⁶
(5³)³ = 5⁹
(5⁴)³ = 5¹²
(5⁵)³ = 5¹⁵

Right column (use pattern):

(4²)⁴ = 4^(2×4) = 4⁸
(6³)⁸ = 6^(3×8) = 6²⁴
(9⁻⁵)⁶ = 9^(-5×6) = 9⁻³⁰
(11⁻²)⁷ = 11^((-2)×(-7)) = 11¹⁴

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Final Answer:

Section A:
1) 1
2) -8
3) 1
4) 343
5) -109
6) \( \frac{9}{16} \)

Section B:
2) \( 9^{13} \)
3) \( 2^6 \)
4) \( 8^5 \)
5) \( 7^2 \)
6) \( 12^4 \)
7) \( 4^{20} \)
8) 1
9) \( 2^5 \)
10) \( 6^{-8} \)
11) \( 15^{-7} \)
12) \( 7^{-3} \)

Section C:
1) \( 2^6 \)
2) \( 8^{-3} \)
3) \( 10^{-19} \)
4) \( 3^{-5} \)
5) \( 3^6 \)
6) \( 6^{10} \)
7) \( 7^{-11} \)
8) \( 14^{27} \)
9) \( 4^7 \)
10) \( 6^{-4} \)
11) \( 20^{-1} \)
12) \( e^7 \)

Extension:
Left:
(2²)² = 2⁴
(2³)² = 2⁶
(2⁴)² = 2⁸
(2⁵)² = 2¹⁰

Middle:
(5²)³ = 5⁶
(5³)³ = 5⁹
(5⁴)³ = 5¹²
(5⁵)³ = 5¹⁵

Right:
(4²)⁴ = 4⁸
(6³)⁸ = 6²⁴
(9⁻⁵)⁶ = 9⁻³⁰
(11⁻²)⁻⁷ = 11¹⁴