Question

error analysis:
example 1 (1 error):
\\(\frac{x^7}{x^3}\\)
\\(x^{\frac{7}{3}}\\)

example 2 (1 error):
\\((a^3)^0\\)
\\(a^3\\)

example 3 (1 error):
\\(10x^{-4}\\)
\\(-10x^4\\)

corrected example with annotations:

Explanation:

Example 1:

Step 1: Recall exponent rule for division

When dividing exponents with the same base \( \frac{x^m}{x^n}=x^{m - n} \), not \( x^{\frac{m}{n}} \).

Step 2: Apply the correct rule

For \( \frac{x^7}{x^3} \), using \( x^{m - n} \), we get \( x^{7 - 3}=x^4 \).

Example 2:

Step 1: Recall exponent rule for power of a power

The rule is \( (a^m)^n=a^{m\times n} \), and also \( a^0 = 1\) for \( a
eq0 \).

Step 2: Correct the calculation

First, \( (a^3)^0=a^{3\times0}=a^0 = 1\) (assuming \( a
eq0 \)), not \( a^3 \).

Example 3:

Step 1: Recall negative exponent rule

The rule is \( x^{-n}=\frac{1}{x^n} \), so \( 10x^{-4}=\frac{10}{x^4} \), not \( - 10x^4 \).

Step 2: Write the correct form

\( 10x^{-4}=\frac{10}{x^4} \)

Corrected Examples with Annotations:

Example 1:

• Original Error: Used \( x^{\frac{m}{n}} \) instead of \( x^{m - n} \) for \( \frac{x^m}{x^n} \).
• Correction: \( \frac{x^7}{x^3}=x^{7 - 3}=x^4 \) (By the quotient rule of exponents: \( \frac{x^m}{x^n}=x^{m - n} \) for \( x

eq0 \), \( m,n\in\mathbb{R} \)).

Example 2:

• Original Error: Ignored the zero - exponent rule (\( a^0 = 1\) for \( a

eq0 \)) and misapplied the power - of - a - power rule.

• Correction: \( (a^3)^0=a^{3\times0}=a^0 = 1\) (By the power - of - a - power rule \( (a^m)^n=a^{m\times n} \) and zero - exponent rule \( a^0 = 1\) for \( a

eq0 \)).

Example 3:

• Original Error: Misapplied the negative - exponent rule (\( x^{-n}=\frac{1}{x^n} \), not \( -x^n \)).
• Correction: \( 10x^{-4}=\frac{10}{x^4} \) (By the negative - exponent rule: \( x^{-n}=\frac{1}{x^n} \) for \( x

eq0 \), \( n\in\mathbb{N} \)).

Answer:

Example 1:

Step 1: Recall exponent rule for division

When dividing exponents with the same base \( \frac{x^m}{x^n}=x^{m - n} \), not \( x^{\frac{m}{n}} \).

Step 2: Apply the correct rule

For \( \frac{x^7}{x^3} \), using \( x^{m - n} \), we get \( x^{7 - 3}=x^4 \).

Example 2:

Step 1: Recall exponent rule for power of a power

The rule is \( (a^m)^n=a^{m\times n} \), and also \( a^0 = 1\) for \( a
eq0 \).

Step 2: Correct the calculation

First, \( (a^3)^0=a^{3\times0}=a^0 = 1\) (assuming \( a
eq0 \)), not \( a^3 \).

Example 3:

Step 1: Recall negative exponent rule

The rule is \( x^{-n}=\frac{1}{x^n} \), so \( 10x^{-4}=\frac{10}{x^4} \), not \( - 10x^4 \).

Step 2: Write the correct form

\( 10x^{-4}=\frac{10}{x^4} \)

Corrected Examples with Annotations:

Example 1:

• Original Error: Used \( x^{\frac{m}{n}} \) instead of \( x^{m - n} \) for \( \frac{x^m}{x^n} \).
• Correction: \( \frac{x^7}{x^3}=x^{7 - 3}=x^4 \) (By the quotient rule of exponents: \( \frac{x^m}{x^n}=x^{m - n} \) for \( x

eq0 \), \( m,n\in\mathbb{R} \)).

Example 2:

• Original Error: Ignored the zero - exponent rule (\( a^0 = 1\) for \( a

eq0 \)) and misapplied the power - of - a - power rule.

• Correction: \( (a^3)^0=a^{3\times0}=a^0 = 1\) (By the power - of - a - power rule \( (a^m)^n=a^{m\times n} \) and zero - exponent rule \( a^0 = 1\) for \( a

eq0 \)).

Example 3:

• Original Error: Misapplied the negative - exponent rule (\( x^{-n}=\frac{1}{x^n} \), not \( -x^n \)).
• Correction: \( 10x^{-4}=\frac{10}{x^4} \) (By the negative - exponent rule: \( x^{-n}=\frac{1}{x^n} \) for \( x

eq0 \), \( n\in\mathbb{N} \)).