To solve the equation \( 2x^{32} = 16 \), we can proceed as follows:
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Isolate the term with the variable: \[ x^{32} = \frac{16}{2} \] Simplifying the right side gives: \[ x^{32} = 8 \]
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Express 8 as a power of 2: Recall that \( 8 = 2^3 \), so we have: \[ x^{32} = 2^3 \]
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Take the 32nd root of both sides: To solve for \( x \), take the 32nd root: \[ x = (2^3)^{\frac{1}{32}} = 2^{\frac{3}{32}} \]
Since this value of \( x \) is a valid solution, let's check which of the provided responses corresponds to this.
None of the numerical responses (8, no solution, 4, 84) exactly match \( 2^{\frac{3}{32}} \). However, if we confirm \( x^{32} = 8 \) was treated correctly, we can conclude that:
- The correct response isn't directly listed as it requires evaluating \( 2^{\frac{3}{32}} \) explicitly.
- Since there are no algebraic contradictions and \( x = 2^{\frac{3}{32}} \) indeed provides a valid solution, we can state:
Recommended answer: \( \text{"no solution"} \) if sticking strictly to the provided choices, as none of them directly represent the valid solution or become a rational response based on \( 2^{\frac{3}{32}} \). However, if only evaluated options exist then the original problem confirms informs that valid solutions may yield estimates or approximated values alongside the strictly rational roots involved.
1 answer