Question
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Tutor-Verified Answer
Answer
The solutions are \(a=-48\) and \(a=16\).
Solution
Solve the equation \( 16=(a+16)^{\frac{4}{5}} \).
Solve the equation by following steps:
- step0: Solve for \(a\):
\(16=\left(a+16\right)^{\frac{4}{5}}\)
- step1: Swap the sides:
\(\left(a+16\right)^{\frac{4}{5}}=16\)
- step2: Simplify the expression:
\(a+16=\pm 16^{\frac{5}{4}}\)
- step3: Simplify:
\(a+16=\pm 32\)
- step4: Separate into possible cases:
\(\begin{align}&a+16=32\\&a+16=-32\end{align}\)
- step5: Calculate:
\(\begin{align}&a=16\\&a=-48\end{align}\)
- step6: Rewrite:
\(a_{1}=-48,a_{2}=16\)
The solutions to the equation \(16=(a+16)^{\frac{4}{5}}\) are \(a=-48\) and \(a=16\).
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Simplify this solution Beyond the Answer
To solve the equation \( 16=(a+16)^{\frac{4}{5}} \), let's first eliminate the fractional exponent by raising both sides to the power of \( \frac{5}{4} \). This gives us: \[ 16^{\frac{5}{4}} = a + 16 \] Now we need to calculate \( 16^{\frac{5}{4}} \). Since \( 16 = 2^4 \), we can rewrite it as: \[ (2^4)^{\frac{5}{4}} = 2^{4 \cdot \frac{5}{4}} = 2^5 = 32 \] Now, substituting back into the equation: \[ 32 = a + 16 \] To isolate \( a \), we subtract 16 from both sides: \[ a = 32 - 16 = 16 \] So, the solution to the equation is: \[ \boxed{16} \]
