Solving Exponential Equation \( 4^{8x} = 256 \)
We need to find the value of $x$ satisfying the equation \( 4^{8x} = 256 \).
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Rewrite the equation with a common base.
Recognize that $256$ is a power of $4$. Specifically, $256 = 4^4$. Substituting this into the original equation gives: \( 4^{8x} = 4^4 \)
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Equate the exponents.
Since the bases on both sides of the equation are the same ($4$), the exponents must be equal: \( 8x = 4 \)
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Solve for $x$.
To isolate $x$, divide both sides of the equation by $8$: \( x = \frac{4}{8} \)
Simplify the fraction: \( x = \frac{1}{2} \)
The calculation shows that \( x = \frac{1}{2} \). The provided correct answer is Option C ($\frac{2}{3}$).
Important Questions from Powers and Exponents
If a real variable $x$ satisfies $3^{x^2} = 27 \times 9^x$, then the value of $\frac{2^{x^2}}{(2^{x})^2}$ is:
- What is the value of $\left(\frac{3^{81}}{27^4}\right)^{1/3}$?
The 12 musical notes are given as C, C#, D, D#, E, F, F#, G, G#, A, A#. Frequency of each note is $ \sqrt[12]{2} $ times the frequency of the previous note. If the frequency of the note C is 130.8 Hz, then the ratio of frequencies of notes F# and C is:
- For positive integers $p$ and $q$, with $\frac{p}{q} \neq 1$, $(\frac{p}{q})^{\frac{p}{q}} = p^{(\frac{p}{q}-1)}$. Then,
- If $7^{3x} = 216$, the value of $7^{-x}$ (rounded off to three decimal places) is ________.
