Incorrect exponent and power statements to identify and correct.

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) \( 2^5 = 10 \)

#### 1. Identify the mistake:
The statement claims that \( 2^5 = 10 \), but this is incorrect.

#### 2. Correct calculation:
\[ 2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32 \]

#### 3. Justification:
The correct value of \( 2^5 \) is 32, not 10. The mistake lies in the incorrect computation of the power.

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

#### 1. Identify the mistake:
The statement claims that \( (-2)^3 = 8 \), but this is incorrect.

#### 2. Correct calculation:
\[ (-2)^3 = (-2) \times (-2) \times (-2) = 4 \times (-2) = -8 \]

#### 3. Justification:
The correct value of \( (-2)^3 \) is \(-8\), not 8. The mistake lies in ignoring the negative sign when multiplying three negative numbers (an odd number of negatives results in a negative product).

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

#### 1. Identify the mistake:
The statement claims that \( -6^2 = 36 \), but this is incorrect.

#### 2. Correct calculation:
\[ -6^2 = -(6^2) = -(6 \times 6) = -36 \]

#### 3. Justification:
The expression \( -6^2 \) means "the negative of \( 6^2 \)," not "the square of \(-6\)." Squaring \( 6 \) gives 36, and taking the negative of that gives \(-36\). The mistake lies in misunderstanding the order of operations (exponentiation before negation).

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d) \( x^0 = 0 \)

#### 1. Identify the mistake:
The statement claims that \( x^0 = 0 \), but this is incorrect.

#### 2. Correct calculation:
For any nonzero \( x \), \( x^0 = 1 \). This is a fundamental property of exponents.

#### 3. Justification:
The correct value of \( x^0 \) is 1, not 0. The mistake lies in misunderstanding the definition of \( x^0 \), which is always 1 for any nonzero \( x \).

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

#### 1. Identify the mistake:
The statement claims that \( x^3 \cdot x^4 = x^{12} \), but this is incorrect.

#### 2. Correct calculation:
When multiplying powers with the same base, we add the exponents:
\[ x^3 \cdot x^4 = x^{3+4} = x^7 \]

#### 3. Justification:
The correct result is \( x^7 \), not \( x^{12} \). The mistake lies in incorrectly applying the exponent rule; instead of adding the exponents, the student multiplied them.

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

#### 1. Identify the mistake:
The statement claims that \( \frac{x^{10}}{x^5} = x^2 \), but this is incorrect.

#### 2. Correct calculation:
When dividing powers with the same base, we subtract the exponents:
\[ \frac{x^{10}}{x^5} = x^{10-5} = x^5 \]

#### 3. Justification:
The correct result is \( x^5 \), not \( x^2 \). The mistake lies in incorrectly applying the exponent rule; instead of subtracting the exponents, the student divided them.

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

#### 1. Identify the mistake:
The statement claims that \( (x^3)^5 = x^8 \), but this is incorrect.

#### 2. Correct calculation:
When raising a power to another power, we multiply the exponents:
\[ (x^3)^5 = x^{3 \cdot 5} = x^{15} \]

#### 3. Justification:
The correct result is \( x^{15} \), not \( x^8 \). The mistake lies in incorrectly applying the exponent rule; instead of multiplying the exponents, the student added them.

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

#### 1. Identify the mistake:
The statement claims that \( 7^{-2} = -49 \), but this is incorrect.

#### 2. Correct calculation:
A negative exponent means taking the reciprocal:
\[ 7^{-2} = \frac{1}{7^2} = \frac{1}{49} \]

#### 3. Justification:
The correct value of \( 7^{-2} \) is \( \frac{1}{49} \), not \(-49\). The mistake lies in misunderstanding the meaning of a negative exponent, which does not imply a negative result but rather a reciprocal.

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

#### 1. Identify the mistake:
The statement claims that \( (-3)^4 = -81 \), but this is incorrect.

#### 2. Correct calculation:
\[ (-3)^4 = (-3) \times (-3) \times (-3) \times (-3) = 9 \times 9 = 81 \]

#### 3. Justification:
The correct value of \( (-3)^4 \) is 81, not \(-81\). The mistake lies in ignoring the fact that an even number of negative factors results in a positive product.

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

#### 1. Identify the mistake:
The statement claims that \( \frac{x^5}{x^9} = x^4 \), but this is incorrect.

#### 2. Correct calculation:
When dividing powers with the same base, we subtract the exponents:
\[ \frac{x^5}{x^9} = x^{5-9} = x^{-4} \]

#### 3. Justification:
The correct result is \( x^{-4} \), not \( x^4 \). The mistake lies in incorrectly applying the exponent rule; instead of subtracting the exponents, the student added them.

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Final Answer:

\[
\boxed{
\begin{array}{ll}
\text{a) } 2^5 = 32, \text{ not 10.} \\
\text{b) } (-2)^3 = -8, \text{ not 8.} \\
\text{c) } -6^2 = -36, \text{ not 36.} \\
\text{d) } x^0 = 1, \text{ not 0.} \\
\text{e) } x^3 \cdot x^4 = x^7, \text{ not } x^{12}. \\
\text{f) } \frac{x^{10}}{x^5} = x^5, \text{ not } x^2. \\
\text{g) } (x^3)^5 = x^{15}, \text{ not } x^8. \\
\text{h) } 7^{-2} = \frac{1}{49}, \text{ not } -49. \\
\text{i) } (-3)^4 = 81, \text{ not } -81. \\
\text{j) } \frac{x^5}{x^9} = x^{-4}, \text{ not } x^4.
\end{array}
}
\]