8th Grade Unit 6: Exponents (Part 1 - Exponent Properties) | Count ... - Free Printable

Problem Analysis:

The task requires identifying the mistakes in each of the given statements and justifying why they are incorrect. Let's analyze each statement step by step.

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a) \( 2^3 = 10 \)

#### Identification of Mistake:
The statement claims that \( 2^3 = 10 \). However, this is incorrect because:
\[ 2^3 = 2 \times 2 \times 2 = 8 \]

#### Justification:
The correct value of \( 2^3 \) is 8, not 10. The mistake lies in the calculation of the exponentiation.

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b) \( (-2)^3 = 8 \)

#### Identification of Mistake:
The statement claims that \( (-2)^3 = 8 \). However, this is incorrect because:
\[ (-2)^3 = (-2) \times (-2) \times (-2) = 4 \times (-2) = -8 \]

#### Justification:
When raising a negative number to an odd power, the result remains negative. Here, \( (-2)^3 = -8 \), not 8.

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c) \( -6^2 = 36 \)

#### Identification of Mistake:
The statement claims that \( -6^2 = 36 \). However, this is incorrect because:
\[ -6^2 = -(6^2) = -(6 \times 6) = -36 \]

#### Justification:
The expression \( -6^2 \) means "the negative of \( 6^2 \)," not "negative 6 squared." Squaring 6 gives 36, and taking the negative of that results in \(-36\).

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d) \( x^0 = 0 \)

#### Identification of Mistake:
The statement claims that \( x^0 = 0 \). However, this is incorrect because:
\[ x^0 = 1 \quad \text{for any } x \neq 0 \]

#### Justification:
Any non-zero number raised to the power of 0 is defined as 1. This is a fundamental property of exponents.

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e) \( x^3 \cdot x^4 = x^{12} \)

#### Identification of Mistake:
The statement claims that \( x^3 \cdot x^4 = x^{12} \). However, this is incorrect because:
\[ x^3 \cdot x^4 = x^{3+4} = x^7 \]

#### Justification:
When multiplying powers with the same base, the exponents are added. Here, \( 3 + 4 = 7 \), so the correct result is \( x^7 \), not \( x^{12} \).

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f) \( \frac{x^{10}}{x^5} = x^2 \)

#### Identification of Mistake:
The statement claims that \( \frac{x^{10}}{x^5} = x^2 \). However, this is incorrect because:
\[ \frac{x^{10}}{x^5} = x^{10-5} = x^5 \]

#### Justification:
When dividing powers with the same base, the exponents are subtracted. Here, \( 10 - 5 = 5 \), so the correct result is \( x^5 \), not \( x^2 \).

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g) \( (x^3)^5 = x^8 \)

#### Identification of Mistake:
The statement claims that \( (x^3)^5 = x^8 \). However, this is incorrect because:
\[ (x^3)^5 = x^{3 \cdot 5} = x^{15} \]

#### Justification:
When raising a power to another power, the exponents are multiplied. Here, \( 3 \cdot 5 = 15 \), so the correct result is \( x^{15} \), not \( x^8 \).

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h) \( 7^2 = -49 \)

#### Identification of Mistake:
The statement claims that \( 7^2 = -49 \). However, this is incorrect because:
\[ 7^2 = 7 \times 7 = 49 \]

#### Justification:
Squaring a positive number always results in a positive number. Here, \( 7^2 = 49 \), not \(-49\).

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i) \( (-3)^4 = -81 \)

#### Identification of Mistake:
The statement claims that \( (-3)^4 = -81 \). However, this is incorrect because:
\[ (-3)^4 = (-3) \times (-3) \times (-3) \times (-3) = 9 \times 9 = 81 \]

#### Justification:
When raising a negative number to an even power, the result is positive. Here, \( (-3)^4 = 81 \), not \(-81\).

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j) \( \frac{x^5}{x^9} = x^4 \)

#### Identification of Mistake:
The statement claims that \( \frac{x^5}{x^9} = x^4 \). However, this is incorrect because:
\[ \frac{x^5}{x^9} = x^{5-9} = x^{-4} \]

#### Justification:
When dividing powers with the same base, the exponents are subtracted. Here, \( 5 - 9 = -4 \), so the correct result is \( x^{-4} \), not \( x^4 \).

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Final Answer:

\[
\boxed{
\begin{array}{ll}
\text{a) } 2^3 = 8, \text{ not 10.} \\
\text{b) } (-2)^3 = -8, \text{ not 8.} \\
\text{c) } -6^2 = -36, \text{ not 36.} \\
\text{d) } x^0 = 1, \text{ not 0.} \\
\text{e) } x^3 \cdot x^4 = x^7, \text{ not } x^{12}. \\
\text{f) } \frac{x^{10}}{x^5} = x^5, \text{ not } x^2. \\
\text{g) } (x^3)^5 = x^{15}, \text{ not } x^8. \\
\text{h) } 7^2 = 49, \text{ not } -49. \\
\text{i) } (-3)^4 = 81, \text{ not } -81. \\
\text{j) } \frac{x^5}{x^9} = x^{-4}, \text{ not } x^4.
\end{array}
}
\]