Question
\( \left(x^{3}\right)^{w}=x^{39} \) What value does \( w \) have to be?
Ask by Sanders Tyler. in the United States
Feb 13,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( w = 13 \)
Solution
We start with the equation (x³)^w = x³⁹.
Using the power of a power rule, we simplify the left-hand side:
(x³)^w = x^(3w).
Now we have:
x^(3w) = x^(39).
Assuming x ≠ 0, the exponents must be equal:
3w = 39.
Solving for w:
w = 39 / 3 = 13.
Thus, the value of w is 13.
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Simplify this solution The Deep Dive
To solve for \( w \) in the equation \( \left(x^{3}\right)^{w}=x^{39} \), we can use the power of a power property of exponents, which states that \( \left(a^{m}\right)^{n} = a^{m \cdot n} \). Applying this property, we can rewrite the left side of the equation: \[ \left(x^{3}\right)^{w} = x^{3w} \] Thus, the equation becomes: \[ x^{3w} = x^{39} \] Since the bases are the same, we can set the exponents equal to each other: \[ 3w = 39 \] Now, we solve for \( w \) by dividing both sides by 3: \[ w = \frac{39}{3} = 13 \] So, the value of \( w \) is \( 13 \).
