Which expression is equivalent to [tex]\left(\frac{x^{-4} y}{x^{-9} y^5}\right)^{-2}[/tex]? Assume [tex]x \neq 0, y \neq 0[/tex].

A. [tex]\frac{y^8}{x^{10}}[/tex]

B. [tex]\frac{x^5}{y^7}[/tex]

C. [tex]\frac{x^5}{y^4}[/tex]

D. [tex]\frac{x}{y^7}[/tex]



Answer :

To determine which expression is equivalent to [tex]\(\left(\frac{x^{-4} y}{x^{-9} y^5}\right)^{-2}\)[/tex], we will simplify the given expression step-by-step.

### Step 1: Simplify the Inner Fraction

Starting with the inner fraction:
[tex]\[ \frac{x^{-4} y}{x^{-9} y^5} \][/tex]

We can handle each component separately:

#### Simplify the [tex]\(x\)[/tex] terms:
[tex]\[ \frac{x^{-4}}{x^{-9}} = x^{-4 - (-9)} = x^{-4 + 9} = x^5 \][/tex]

#### Simplify the [tex]\(y\)[/tex] terms:
[tex]\[ \frac{y}{y^5} = y^{1 - 5} = y^{-4} \][/tex]

Combining these simplified components, the inner fraction becomes:
[tex]\[ \frac{x^{-4} y}{x^{-9} y^5} = \frac{x^5}{y^4} \][/tex]

### Step 2: Apply the Outer Exponent

Next, we need to apply the exponent [tex]\(-2\)[/tex] to the simplified inner fraction:
[tex]\[ \left(\frac{x^5}{y^4}\right)^{-2} \][/tex]

Using the property of exponents [tex]\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)[/tex], we get:
[tex]\[ \left(\frac{x^5}{y^4}\right)^{-2} = \frac{(x^5)^{-2}}{(y^4)^{-2}} \][/tex]

### Step 3: Simplify the Resulting Expression

Now, simplify the exponents:
[tex]\[ (x^5)^{-2} = x^{5 \cdot (-2)} = x^{-10} \][/tex]
[tex]\[ (y^4)^{-2} = y^{4 \cdot (-2)} = y^{-8} \][/tex]

So, we have:
[tex]\[ \frac{x^{-10}}{y^{-8}} \][/tex]

### Step 4: Rewrite with Positive Exponents

To write this expression with positive exponents, we use the property [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex]:
[tex]\[ \frac{x^{-10}}{y^{-8}} = x^{-10} \cdot y^8 = \frac{y^8}{x^{10}} \][/tex]

Thus, the equivalent expression we were looking for is:
[tex]\[ \boxed{\frac{y^8}{x^{10}}} \][/tex]

This matches the first option in the provided list, confirming that:

[tex]\[ \left(\frac{x^{-4} y}{x^{-9} y^5}\right)^{-2} \quad \text{is equivalent to} \quad \frac{y^8}{x^{10}}. \][/tex]