8th Grade Common Core Math Worksheets - Free Printable

Problem: Simplify Expressions Involving Multiplication of Exponents

The task involves simplifying expressions using the properties of exponents. Let's solve each problem step by step.

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#### 1. The expression \(2^3 \cdot 4^2\) is equivalent to:
- Given: \(2^3 \cdot 4^2\)
- Step 1: Rewrite \(4\) as a power of \(2\): \(4 = 2^2\).
\[
4^2 = (2^2)^2 = 2^{2 \cdot 2} = 2^4
\]
- Step 2: Substitute back into the expression:
\[
2^3 \cdot 4^2 = 2^3 \cdot 2^4
\]
- Step 3: Use the property of exponents \(a^m \cdot a^n = a^{m+n}\):
\[
2^3 \cdot 2^4 = 2^{3+4} = 2^7
\]
- Answer: \(2^7\)

Correct Choice: 1) \(2^7\)

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#### 2. Which expression is equivalent to \(3^3 \cdot 3^4\)?
- Given: \(3^3 \cdot 3^4\)
- Step 1: Use the property of exponents \(a^m \cdot a^n = a^{m+n}\):
\[
3^3 \cdot 3^4 = 3^{3+4} = 3^7
\]
- Answer: \(3^7\)

Correct Choice: 4) \(3^7\)

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#### 3. The expression \(3^2 \cdot 3^3 \cdot 3^4\) is equivalent to:
- Given: \(3^2 \cdot 3^3 \cdot 3^4\)
- Step 1: Use the property of exponents \(a^m \cdot a^n \cdot a^p = a^{m+n+p}\):
\[
3^2 \cdot 3^3 \cdot 3^4 = 3^{2+3+4} = 3^9
\]
- Answer: \(3^9\)

Correct Choice: 3) \(3^9\)

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#### 4. The product of \(3x^5\) and \(2x^4\) is:
- Given: \(3x^5 \cdot 2x^4\)
- Step 1: Multiply the coefficients:
\[
3 \cdot 2 = 6
\]
- Step 2: Use the property of exponents \(x^m \cdot x^n = x^{m+n}\) for the variable part:
\[
x^5 \cdot x^4 = x^{5+4} = x^9
\]
- Step 3: Combine the results:
\[
3x^5 \cdot 2x^4 = 6x^9
\]
- Answer: \(6x^9\)

Correct Choice: 3) \(6x^9\)

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#### 5. The product of \(2x^3\) and \(6x^4\) is:
- Given: \(2x^3 \cdot 6x^4\)
- Step 1: Multiply the coefficients:
\[
2 \cdot 6 = 12
\]
- Step 2: Use the property of exponents \(x^m \cdot x^n = x^{m+n}\) for the variable part:
\[
x^3 \cdot x^4 = x^{3+4} = x^7
\]
- Step 3: Combine the results:
\[
2x^3 \cdot 6x^4 = 12x^7
\]
- Answer: \(12x^7\)

Correct Choice: 2) \(12x^8\) (Note: There seems to be a typo in the options; it should be \(12x^7\).)

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#### 6. Which expression represents \((3x^2y^4)(4xy^2)\) in simplest form?
- Given: \((3x^2y^4)(4xy^2)\)
- Step 1: Multiply the coefficients:
\[
3 \cdot 4 = 12
\]
- Step 2: Use the property of exponents \(x^m \cdot x^n = x^{m+n}\) for the \(x\) terms:
\[
x^2 \cdot x = x^{2+1} = x^3
\]
- Step 3: Use the property of exponents \(y^m \cdot y^n = y^{m+n}\) for the \(y\) terms:
\[
y^4 \cdot y^2 = y^{4+2} = y^6
\]
- Step 4: Combine the results:
\[
(3x^2y^4)(4xy^2) = 12x^3y^6
\]
- Answer: \(12x^3y^6\)

Correct Choice: 4) \(12x^3y^6\)

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#### 7. The product of \(4x^2y\) and \(2xy^3\) is:
- Given: \(4x^2y \cdot 2xy^3\)
- Step 1: Multiply the coefficients:
\[
4 \cdot 2 = 8
\]
- Step 2: Use the property of exponents \(x^m \cdot x^n = x^{m+n}\) for the \(x\) terms:
\[
x^2 \cdot x = x^{2+1} = x^3
\]
- Step 3: Use the property of exponents \(y^m \cdot y^n = y^{m+n}\) for the \(y\) terms:
\[
y \cdot y^3 = y^{1+3} = y^4
\]
- Step 4: Combine the results:
\[
4x^2y \cdot 2xy^3 = 8x^3y^4
\]
- Answer: \(8x^3y^4\)

Correct Choice: 3) \(8x^3y^4\)

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#### 8. The product of \(6x^3y^3\) and \(2x^2y\) is:
- Given: \(6x^3y^3 \cdot 2x^2y\)
- Step 1: Multiply the coefficients:
\[
6 \cdot 2 = 12
\]
- Step 2: Use the property of exponents \(x^m \cdot x^n = x^{m+n}\) for the \(x\) terms:
\[
x^3 \cdot x^2 = x^{3+2} = x^5
\]
- Step 3: Use the property of exponents \(y^m \cdot y^n = y^{m+n}\) for the \(y\) terms:
\[
y^3 \cdot y = y^{3+1} = y^4
\]
- Step 4: Combine the results:
\[
6x^3y^3 \cdot 2x^2y = 12x^5y^4
\]
- Answer: \(12x^5y^4\)

Correct Choice: 3) \(12x^5y^4\)

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#### 9. The expression \((x^2z^3)(xy^2z)\) is equivalent to:
- Given: \((x^2z^3)(xy^2z)\)
- Step 1: Multiply the coefficients (there are no numerical coefficients, so we proceed with variables):
- Step 2: Use the property of exponents \(x^m \cdot x^n = x^{m+n}\) for the \(x\) terms:
\[
x^2 \cdot x = x^{2+1} = x^3
\]
- Step 3: Use the property of exponents \(y^m \cdot y^n = y^{m+n}\) for the \(y\) terms:
\[
y^0 \cdot y^2 = y^{0+2} = y^2 \quad (\text{since } y^0 = 1)
\]
- Step 4: Use the property of exponents \(z^m \cdot z^n = z^{m+n}\) for the \(z\) terms:
\[
z^3 \cdot z = z^{3+1} = z^4
\]
- Step 5: Combine the results:
\[
(x^2z^3)(xy^2z) = x^3y^2z^4
\]
- Answer: \(x^3y^2z^4\)

Correct Choice: 2) \(x^3y^2z^4\)

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Final Answers:

1. \(2^7\)
2. \(3^7\)
3. \(3^9\)
4. \(6x^9\)
5. \(12x^7\)
6. \(12x^3y^6\)
7. \(8x^3y^4\)
8. \(12x^5y^4\)
9. \(x^3y^2z^4\)

\[
\boxed{1, 4, 3, 3, 2, 4, 3, 3, 2}
\]