Eighth Grade Exponent Rules Practice and Color by Number - Free Printable

Problem Description:

The image shows a worksheet titled "Exponent Rules Practice and Color by Number". The task involves simplifying expressions using exponent rules and then coloring a mandala based on the answers. Below is the breakdown of the problem and its solution.

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Exponent Rules Recap:

1. Multiplying Powers with the Same Base:
\( a^m \cdot a^n = a^{m+n} \)
(Add the exponents when multiplying powers with the same base.)

2. Dividing Powers with the Same Base:
\( \frac{a^m}{a^n} = a^{m-n} \)
(Subtract the exponents when dividing powers with the same base.)

3. Power of a Power:
\( (a^m)^n = a^{m \cdot n} \)
(Multiply the exponents when raising a power to another power.)

4. Power of a Product:
\( (ab)^m = a^m \cdot b^m \)
(Distribute the exponent to each factor in the product.)

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Worksheet Problems:

The worksheet provides several expressions to simplify. Here are the problems and their solutions:

#### 1. Simplify: \( x^3 \cdot x^5 \)
- Rule: Multiplying Powers with the Same Base (\( a^m \cdot a^n = a^{m+n} \))
- Solution:
\[
x^3 \cdot x^5 = x^{3+5} = x^8
\]
- Answer: \( x^8 \)

#### 2. Simplify: \( \frac{y^7}{y^3} \)
- Rule: Dividing Powers with the Same Base (\( \frac{a^m}{a^n} = a^{m-n} \))
- Solution:
\[
\frac{y^7}{y^3} = y^{7-3} = y^4
\]
- Answer: \( y^4 \)

#### 3. Simplify: \( (z^2)^4 \)
- Rule: Power of a Power (\( (a^m)^n = a^{m \cdot n} \))
- Solution:
\[
(z^2)^4 = z^{2 \cdot 4} = z^8
\]
- Answer: \( z^8 \)

#### 4. Simplify: \( (2x)^3 \)
- Rule: Power of a Product (\( (ab)^m = a^m \cdot b^m \))
- Solution:
\[
(2x)^3 = 2^3 \cdot x^3 = 8x^3
\]
- Answer: \( 8x^3 \)

#### 5. Simplify: \( \frac{a^6}{a^2} \)
- Rule: Dividing Powers with the Same Base (\( \frac{a^m}{a^n} = a^{m-n} \))
- Solution:
\[
\frac{a^6}{a^2} = a^{6-2} = a^4
\]
- Answer: \( a^4 \)

#### 6. Simplify: \( (b^3)^2 \)
- Rule: Power of a Power (\( (a^m)^n = a^{m \cdot n} \))
- Solution:
\[
(b^3)^2 = b^{3 \cdot 2} = b^6
\]
- Answer: \( b^6 \)

#### 7. Simplify: \( (3c)^2 \)
- Rule: Power of a Product (\( (ab)^m = a^m \cdot b^m \))
- Solution:
\[
(3c)^2 = 3^2 \cdot c^2 = 9c^2
\]
- Answer: \( 9c^2 \)

#### 8. Simplify: \( x^2 \cdot x^4 \)
- Rule: Multiplying Powers with the Same Base (\( a^m \cdot a^n = a^{m+n} \))
- Solution:
\[
x^2 \cdot x^4 = x^{2+4} = x^6
\]
- Answer: \( x^6 \)

#### 9. Simplify: \( \frac{m^5}{m} \)
- Rule: Dividing Powers with the Same Base (\( \frac{a^m}{a^n} = a^{m-n} \))
- Solution:
\[
\frac{m^5}{m} = m^{5-1} = m^4
\]
- Answer: \( m^4 \)

#### 10. Simplify: \( (d^2)^3 \)
- Rule: Power of a Power (\( (a^m)^n = a^{m \cdot n} \))
- Solution:
\[
(d^2)^3 = d^{2 \cdot 3} = d^6
\]
- Answer: \( d^6 \)

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Coloring Instructions:

After simplifying each expression, the student should match the simplified answer to the corresponding color code provided on the worksheet. For example:
- If \( x^8 \) corresponds to "pink," then all sections labeled with \( x^8 \) in the mandala should be colored pink.
- Repeat this process for all answers.

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Final Answer:

The simplified expressions are:
1. \( x^8 \)
2. \( y^4 \)
3. \( z^8 \)
4. \( 8x^3 \)
5. \( a^4 \)
6. \( b^6 \)
7. \( 9c^2 \)
8. \( x^6 \)
9. \( m^4 \)
10. \( d^6 \)

\[
\boxed{x^8, y^4, z^8, 8x^3, a^4, b^6, 9c^2, x^6, m^4, d^6}
\]

Use these answers to color the mandala according to the provided color key.