Question

x^{-4}
-4 / x
-4x
x^4
1 / (x^4)
question 8 1 pts
(2x^2)^3
expand this out to help
2x^5
8x^6
8x^5
2x^6

Explanation:

First Question (Simplify \( x^{-4} \))

Step1: Recall negative exponent rule

The rule for negative exponents is \( a^{-n} = \frac{1}{a^n} \) (where \( a
eq 0 \) and \( n \) is a positive integer). For \( x^{-4} \), we apply this rule with \( a = x \) and \( n = 4 \).

Step2: Apply the rule

Using \( a^{-n} = \frac{1}{a^n} \), we substitute \( a = x \) and \( n = 4 \), so \( x^{-4} = \frac{1}{x^4} \).

Step1: Recall power of a product rule

The power of a product rule states that \( (ab)^n = a^n b^n \). For \( (2x^2)^3 \), we apply this rule with \( a = 2 \), \( b = x^2 \), and \( n = 3 \).

Step2: Apply the rule to each factor

First, handle the coefficient: \( 2^3 = 8 \). Then, handle the variable part using the power of a power rule \( (x^m)^n = x^{m \cdot n} \). For \( (x^2)^3 \), \( m = 2 \) and \( n = 3 \), so \( (x^2)^3 = x^{2 \cdot 3} = x^6 \).

Step3: Combine the results

Multiply the coefficient result and the variable result: \( 2^3 \cdot (x^2)^3 = 8 \cdot x^6 = 8x^6 \).

Answer:

D. \( 1 / (x^4) \)

Second Question (Expand \( (2x^2)^3 \))