Question
1. Identify the value of the product of the exponential power on the \( x \) and \( y \) terms. \( \frac{12 x^{5} z^{8} y^{9}}{2 x^{-1} z^{4} y^{4}} \)
Ask by Cummings Harris. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The product of the exponents on the \( x \) and \( y \) terms is 30.
Solution
We start with the expression
(12 x⁵ z⁸ y⁹) / (2 x⁻¹ z⁴ y⁴).
Step 1. Simplify the numerical coefficient:
12/2 = 6.
Step 2. Simplify the x‐terms:
x⁵ / x⁻¹ = x^(5 - (–1)) = x^(5 + 1) = x⁶.
Step 3. Simplify the z‐terms:
z⁸ / z⁴ = z^(8 – 4) = z⁴.
Step 4. Simplify the y‐terms:
y⁹ / y⁴ = y^(9 – 4) = y⁵.
Thus, the entire expression becomes:
6 x⁶ z⁴ y⁵.
Since the problem asks for "the product of the exponential power on the x and y terms," we focus on the exponents attached to x and y. Here, x has an exponent of 6 and y has an exponent of 5. Their product (multiplying the exponents) is:
6 × 5 = 30.
So, the answer is 30.
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Simplify this solution Beyond the Answer
To find the product of the exponential powers on the \(x\) and \(y\) terms, we first simplify the given expression: \[ \frac{12 x^{5} z^{8} y^{9}}{2 x^{-1} z^{4} y^{4}} = \frac{12}{2} \cdot \frac{x^{5}}{x^{-1}} \cdot \frac{z^{8}}{z^{4}} \cdot \frac{y^{9}}{y^{4}} \] This simplifies to: \[ 6 \cdot x^{5 - (-1)} \cdot z^{8 - 4} \cdot y^{9 - 4} = 6 \cdot x^{6} \cdot z^{4} \cdot y^{5} \] Now, the value of the product of the exponential powers for \(x\) and \(y\) is: - For \(x\), the power is \(6\). - For \(y\), the power is \(5\). So, the product of the powers is: \[ 6 \cdot 5 = 30 \] Therefore, the value of the product of the exponential powers is \(30\).
