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

Let's solve each section of this Exponential Expressions: Multiplying and Dividing worksheet step by step, using the laws of exponents.

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✔ Section A: Find the values

We'll evaluate each expression directly.

1) $ 3^0 = 1 $
> Any non-zero number raised to the power of 0 is 1.

2) $ (-2)^3 = (-2) \times (-2) \times (-2) = -8 $
> Odd power of a negative number is negative.

3) $ (-1)^{20} = 1 $
> Even power of $-1$ is always 1.

4) $ 7^3 = 7 \times 7 \times 7 = 343 $

5) $ (-5)^3 + (-4)^2 = (-125) + (16) = -109 $
> $(-5)^3 = -125$, $(-4)^2 = 16$

6) $ \left(\frac{3}{4}\right)^2 = \frac{3^2}{4^2} = \frac{9}{16} $

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✔ Section B: Simplify

Use exponent rules:
- $ a^m \times a^n = a^{m+n} $
- $ a^m \div a^n = a^{m-n} $
- $ a^0 = 1 $

1) $ 5^2 \times 5^4 = 5^{2+4} = 5^6 $ ✔ (already given)

2) $ 9^{11} \times 9^2 = 9^{11+2} = 9^{13} $

3) $ 2^5 \times 2 = 2^5 \times 2^1 = 2^{5+1} = 2^6 $

4) $ 8^5 \times 8^0 = 8^5 \times 1 = 8^5 $

5) $ 7^5 \div 7^3 = 7^{5-3} = 7^2 $

6) $ 12^{10} \div 12^6 = 12^{10-6} = 12^4 $

7) $ 4^{21} \div 4 = 4^{21} \div 4^1 = 4^{20} $

8) $ 5^3 \div 5^3 = 5^{3-3} = 5^0 = 1 $

9) $ 2^{-3} \times 2^8 = 2^{-3+8} = 2^5 $

10) $ 6^4 \div 6^{12} = 6^{4-12} = 6^{-8} $

11) $ 15^2 \times 15^{-9} = 15^{2-9} = 15^{-7} $

12) $ 7^{10} \div 7^{14} \times 7 = 7^{10-14} \times 7^1 = 7^{-4} \times 7^1 = 7^{-3} $

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✔ Section C: Simplify

Now we simplify expressions involving fractions and negative exponents.

Rules:
- $ \frac{a^m}{a^n} = a^{m-n} $
- $ \frac{a^m \times a^n}{a^p} = a^{m+n-p} $
- $ a^{-n} = \frac{1}{a^n} $

1) $ \frac{2^{10}}{2^4} = 2^{10-4} = 2^6 $

2) $ \frac{8^{-1}}{8^2} = 8^{-1-2} = 8^{-3} $

3) $ \frac{10^{-12}}{10^7} = 10^{-12-7} = 10^{-19} $

4) $ \frac{3^{-6}}{3^{-1}} = 3^{-6 - (-1)} = 3^{-5} $

5) $ \frac{3^3 \times 3^5}{3^2} = \frac{3^{8}}{3^2} = 3^{8-2} = 3^6 $

6) $ \frac{6^{-2} \times 6^4}{6^{-8}} = \frac{6^{2}}{6^{-8}} = 6^{2 - (-8)} = 6^{10} $

7) $ \frac{7^{-8} \times 7^{-7}}{7^{-4}} = \frac{7^{-15}}{7^{-4}} = 7^{-15 - (-4)} = 7^{-11} $

8) $ \frac{14^{-3}}{14^{-11} \times 14^{-19}} = \frac{14^{-3}}{14^{-30}} = 14^{-3 - (-30)} = 14^{27} $

9) $ \frac{4^8 \times 4^5}{4^2 \times 4^4} = \frac{4^{13}}{4^6} = 4^{13-6} = 4^7 $

10) $ \frac{6^{-7} \times 6^3}{6^{-1} \times 6} = \frac{6^{-4}}{6^{0}} = 6^{-4} $

11) $ \frac{20^{-17} \times 20^9}{20^4 \times 20^{-11}} = \frac{20^{-8}}{20^{-7}} = 20^{-8 - (-7)} = 20^{-1} $

12) $ \frac{5e^8 \times 4e^5}{2e^2 \times 10e^4} $
First, group constants and variables:

Numerator: $ 5 \times 4 \times e^{8+5} = 20e^{13} $
Denominator: $ 2 \times 10 \times e^{2+4} = 20e^6 $
So: $ \frac{20e^{13}}{20e^6} = e^{13-6} = e^7 $

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✔ Extension: Investigate the following

We're asked to use patterns to simplify powers of powers.

Recall: $ (a^m)^n = a^{m \times n} $

#### First column:
- $ (2^1)^2 = 2^{1 \times 2} = 2^2 $
- $ (2^2)^2 = 2^{2 \times 2} = 2^4 $
- $ (2^3)^2 = 2^{3 \times 2} = 2^6 $
- $ (2^4)^2 = 2^{4 \times 2} = 2^8 $
- $ (2^5)^2 = 2^{5 \times 2} = 2^{10} $

#### Second column:
- $ (5^1)^3 = 5^{1 \times 3} = 5^3 $
- $ (5^2)^3 = 5^{2 \times 3} = 5^6 $
- $ (5^3)^3 = 5^{3 \times 3} = 5^9 $
- $ (5^4)^3 = 5^{4 \times 3} = 5^{12} $
- $ (5^5)^3 = 5^{5 \times 3} = 5^{15} $

#### Third column:
- $ (4^2)^4 = 4^{2 \times 4} = 4^8 $
- $ (6^3)^8 = 6^{3 \times 8} = 6^{24} $
- $ (9^{-5})^6 = 9^{-5 \times 6} = 9^{-30} $
- $ (11^{-2})^{-7} = 11^{(-2) \times (-7)} = 11^{14} $

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✔ Final Answers Summary

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

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#### Section B
1) $ 5^6 $
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} $

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#### 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 $

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#### Extension
(Left Column)
- $ (2^2)^2 = 2^4 $
- $ (2^3)^2 = 2^6 $
- $ (2^4)^2 = 2^8 $
- $ (2^5)^2 = 2^{10} $

(Middle Column)
- $ (5^2)^3 = 5^6 $
- $ (5^3)^3 = 5^9 $
- $ (5^4)^3 = 5^{12} $
- $ (5^5)^3 = 5^{15} $

(Right Column)
- $ (4^2)^4 = 4^8 $
- $ (6^3)^8 = 6^{24} $
- $ (9^{-5})^6 = 9^{-30} $
- $ (11^{-2})^{-7} = 11^{14} $

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✔ Key Rule Used:
For any base $ a $ and integers $ m, n $:
- $ (a^m)^n = a^{m \times n} $
- $ a^m \times a^n = a^{m+n} $
- $ a^m \div a^n = a^{m-n} $

This worksheet reinforces understanding of exponential laws through practice and pattern recognition. Let me know if you'd like a printable version or further explanation!