Question

question
select the equivalent expression.
\\(\frac{x^{-4} \cdot x^{-5}}{x^{-6}}\\)
answer
\\(\circ\\) \\(x^3\\) \\(\circ\\) \\(\frac{1}{x^7}\\) \\(\circ\\) \\(x^7\\) \\(\circ\\) \\(\frac{1}{x^3}\\)

Explanation:

Step1: Simplify numerator (product rule)

For \( x^a \cdot x^b = x^{a + b} \), so \( x^{-4} \cdot x^{-5} = x^{-4 + (-5)} = x^{-9} \).

Step2: Simplify fraction (quotient rule)

For \( \frac{x^a}{x^b} = x^{a - b} \), so \( \frac{x^{-9}}{x^{-6}} = x^{-9 - (-6)} = x^{-3} \).

Step3: Rewrite negative exponent

\( x^{-3} = \frac{1}{x^3} \)? Wait, no, wait: Wait, -9 - (-6) is -9 + 6 = -3? Wait, no, wait the options have \( \frac{1}{x^3} \)? Wait no, the options: let's check again. Wait the original problem: \( \frac{x^{-4} \cdot x^{-5}}{x^{-6}} \). Step1: numerator: \( x^{-4 + (-5)} = x^{-9} \). Step2: divide by \( x^{-6} \): \( x^{-9 - (-6)} = x^{-9 + 6} = x^{-3} \). But \( x^{-3} = \frac{1}{x^3} \), but the options: wait the options are \( x^3 \), \( \frac{1}{x^7} \), \( x^7 \), \( \frac{1}{x^3} \)? Wait maybe I made a mistake. Wait no, let's re-express:

Wait, \( x^{-4} \cdot x^{-5} = x^{-4 -5} = x^{-9} \). Then divide by \( x^{-6} \): \( x^{-9} \div x^{-6} = x^{-9 - (-6)} = x^{-3} = \frac{1}{x^3} \). Wait but the options: looking at the image, the options are:

First row: left \( x^3 \), right \( x^7 \)

Second row: left \( \frac{1}{x^7} \), right \( \frac{1}{x^3} \)

Ah, so the correct answer is the right option in the second row, \( \frac{1}{x^3} \). Wait, but let's check again. Wait, maybe I messed up the exponent rules. Wait, \( a^m \cdot a^n = a^{m + n} \), correct. \( a^m / a^n = a^{m - n} \), correct. So \( x^{-4} \cdot x^{-5} = x^{-9} \). Then \( x^{-9} / x^{-6} = x^{-9 + 6} = x^{-3} = 1/x^3 \). So the correct option is the fourth one, \( \frac{1}{x^3} \).

Answer:

\(\boldsymbol{\frac{1}{x^3}}\) (the right option in the second row of the answer choices)