Let's simplify the given expression step-by-step and complete the statements accordingly:
Given expression:
[tex]\[
\left[\frac{\left(x^2 y^3\right)^{-1}}{\left(x^{-2} y^2 z\right)^2}\right], \quad x \neq 0, y \neq 0, z \neq 0
\][/tex]
### Step 1: Simplify the Numerator
The numerator is:
[tex]\[
(x^2 y^3)^{-1}
\][/tex]
Using the property [tex]\((a^m)^{-1} = a^{-m}\)[/tex], we get:
[tex]\[
(x^2 y^3)^{-1} = x^{-2} y^{-3}
\][/tex]
### Step 2: Simplify the Denominator
The denominator is:
[tex]\[
(x^{-2} y^2 z)^2
\][/tex]
Using the property [tex]\((a \cdot b \cdot c)^m = a^m \cdot b^m \cdot c^m\)[/tex], we get:
[tex]\[
(x^{-2} y^2 z)^2 = x^{-4} y^4 z^2
\][/tex]
### Step 3: Combine the Numerator and Denominator
We now have the simplified expression as:
[tex]\[
\frac{x^{-2} y^{-3}}{x^{-4} y^4 z^2}
\][/tex]
### Step 4: Simplify the Expression by Subtracting Exponents
When you divide terms with the same base, you subtract their exponents:
[tex]\[
\frac{x^{-2}}{x^{-4}} = x^{(-2) - (-4)} = x^{2}
\][/tex]
[tex]\[
\frac{y^{-3}}{y^4} = y^{-3 - 4} = y^{-7}
\][/tex]
[tex]\[
\frac{1}{z^2} = z^{-2}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
x^2 y^{-7} z^{-2}
\][/tex]
### Step 5: Rewrite the Expression with Positive Exponents
For positive exponents, we move [tex]\( y^{-7} \)[/tex] and [tex]\( z^{-2} \)[/tex] to the denominator:
[tex]\[
x^2 y^{-7} z^{-2} = \frac{x^2}{y^7 z^2}
\][/tex]
### Exponents and Positions
1. The exponent on [tex]\( x \)[/tex] is [tex]\( 2 \)[/tex]. It is in the numerator.
2. The exponent on [tex]\( y \)[/tex] is [tex]\( 7 \)[/tex]. It is in the denominator.
3. The exponent on [tex]\( z \)[/tex] is [tex]\( 2 \)[/tex]. It is in the denominator.
Now, let's complete the statements:
The exponent on [tex]\( x \)[/tex] is [tex]\( 2 \)[/tex] [tex]\(\square\)[/tex] [tex]\(\checkmark\)[/tex].
The exponent on [tex]\( y \)[/tex] is [tex]\( 7 \)[/tex] [tex]\(14\)[/tex] [tex]\(\square\)[/tex] [tex]\(\checkmark\)[/tex].
The exponent on [tex]\( z \)[/tex] is [tex]\( 2 \)[/tex] [tex]\(\square\)[/tex] [tex]\(\checkmark\)[/tex].
[tex]\( x^2 \)[/tex] is in the [tex]\(\text{numerator}\)[/tex].
[tex]\( y^7 \)[/tex] is in the [tex]\(\text{denominator}\)[/tex].
[tex]\( z^2 \)[/tex] is in the [tex]\(\text{denominator}\)[/tex].
So, the final answer is:
- The exponent on [tex]\( x \)[/tex] is [tex]\( 2 \)[/tex].
- The exponent on [tex]\( y \)[/tex] is [tex]\( 7 \)[/tex].
- The exponent on [tex]\( z \)[/tex] is [tex]\( 2 \)[/tex].
- [tex]\( x^2 \)[/tex] is in the numerator.
- [tex]\( y^7 \)[/tex] is in the denominator.
- [tex]\( z^2 \)[/tex] is in the denominator.
