First Law of Exponents Worksheet for simplifying expressions using the product rule.

Problem: Simplify the given expressions using the First Law of Exponents.

The First Law of Exponents states:
\[
a^m \times a^n = a^{m+n}
\]
This rule applies when the bases are the same. We will use this rule to simplify each expression.

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Solution:

#### 1. \( (23)^{2n} \times (23)^{6n} \)
- Both terms have the same base, \( 23 \).
- Using the First Law of Exponents:
\[
(23)^{2n} \times (23)^{6n} = (23)^{2n + 6n} = (23)^{8n}
\]

#### 2. \( (-3)^{49} \times (-3)^{95} \)
- Both terms have the same base, \( -3 \).
- Using the First Law of Exponents:
\[
(-3)^{49} \times (-3)^{95} = (-3)^{49 + 95} = (-3)^{144}
\]

#### 3. \( \left( \frac{2}{3} \right)^4 \times \left( \frac{2}{3} \right)^{-9} \)
- Both terms have the same base, \( \frac{2}{3} \).
- Using the First Law of Exponents:
\[
\left( \frac{2}{3} \right)^4 \times \left( \frac{2}{3} \right)^{-9} = \left( \frac{2}{3} \right)^{4 + (-9)} = \left( \frac{2}{3} \right)^{-5}
\]

#### 4. \( (-13)^0 \times (-13)^{-19} \)
- Recall that any non-zero number raised to the power of 0 is 1:
\[
(-13)^0 = 1
\]
- Using the First Law of Exponents:
\[
(-13)^0 \times (-13)^{-19} = 1 \times (-13)^{-19} = (-13)^{-19}
\]

#### 5. \( \left( -\frac{7}{4} \right)^8 \times \left( -\frac{7}{4} \right)^{-12} \)
- Both terms have the same base, \( -\frac{7}{4} \).
- Using the First Law of Exponents:
\[
\left( -\frac{7}{4} \right)^8 \times \left( -\frac{7}{4} \right)^{-12} = \left( -\frac{7}{4} \right)^{8 + (-12)} = \left( -\frac{7}{4} \right)^{-4}
\]

#### 6. \( \left( \frac{9}{7} \right)^{23} \times \left( \frac{9}{7} \right)^{-97} \)
- Both terms have the same base, \( \frac{9}{7} \).
- Using the First Law of Exponents:
\[
\left( \frac{9}{7} \right)^{23} \times \left( \frac{9}{7} \right)^{-97} = \left( \frac{9}{7} \right)^{23 + (-97)} = \left( \frac{9}{7} \right)^{-74}
\]

#### 7. \( (-9)^{-90} \times (-9)^{-47} \)
- Both terms have the same base, \( -9 \).
- Using the First Law of Exponents:
\[
(-9)^{-90} \times (-9)^{-47} = (-9)^{-90 + (-47)} = (-9)^{-137}
\]

#### 8. \( \left( -\frac{3}{7} \right)^x \times \left( -\frac{3}{7} \right)^2 \)
- Both terms have the same base, \( -\frac{3}{7} \).
- Using the First Law of Exponents:
\[
\left( -\frac{3}{7} \right)^x \times \left( -\frac{3}{7} \right)^2 = \left( -\frac{3}{7} \right)^{x + 2}
\]

#### 9. \( (-22)^{x+7} \times (-22)^7 \)
- Both terms have the same base, \( -22 \).
- Using the First Law of Exponents:
\[
(-22)^{x+7} \times (-22)^7 = (-22)^{(x+7) + 7} = (-22)^{x + 14}
\]

#### 10. \( \left( \frac{x}{2} \right)^{22} \times \left( \frac{x}{2} \right)^{10} \)
- Both terms have the same base, \( \frac{x}{2} \).
- Using the First Law of Exponents:
\[
\left( \frac{x}{2} \right)^{22} \times \left( \frac{x}{2} \right)^{10} = \left( \frac{x}{2} \right)^{22 + 10} = \left( \frac{x}{2} \right)^{32}
\]

#### 11. \( \left( -\frac{7}{6} \right)^8 \times \left( -\frac{7}{6} \right)^{10} \)
- Both terms have the same base, \( -\frac{7}{6} \).
- Using the First Law of Exponents:
\[
\left( -\frac{7}{6} \right)^8 \times \left( -\frac{7}{6} \right)^{10} = \left( -\frac{7}{6} \right)^{8 + 10} = \left( -\frac{7}{6} \right)^{18}
\]

#### 12. \( (-5x)^2 \times (-5x)^3 \)
- Both terms have the same base, \( -5x \).
- Using the First Law of Exponents:
\[
(-5x)^2 \times (-5x)^3 = (-5x)^{2 + 3} = (-5x)^5
\]

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Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ (23)^{8n} \\
2. & \ (-3)^{144} \\
3. & \ \left( \frac{2}{3} \right)^{-5} \\
4. & \ (-13)^{-19} \\
5. & \ \left( -\frac{7}{4} \right)^{-4} \\
6. & \ \left( \frac{9}{7} \right)^{-74} \\
7. & \ (-9)^{-137} \\
8. & \ \left( -\frac{3}{7} \right)^{x+2} \\
9. & \ (-22)^{x+14} \\
10. & \ \left( \frac{x}{2} \right)^{32} \\
11. & \ \left( -\frac{7}{6} \right)^{18} \\
12. & \ (-5x)^5
\end{aligned}
}
\]