Questions: -(4x+1)^(3/2) = 27

-(4x+1)^(3/2) = 27
Transcript text: $-(4 x+1)^{3 / 2}=27$
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Solution

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Solution Steps

To solve the equation \(-(4x + 1)^{3/2} = 27\), we need to isolate \(x\). First, we can eliminate the negative sign by multiplying both sides by -1. Then, we raise both sides to the power of \(2/3\) to get rid of the exponent. Finally, we solve for \(x\).

Step 1: Isolate the Exponential Term

Given the equation: \[ -(4x + 1)^{3/2} = 27 \] First, multiply both sides by \(-1\) to eliminate the negative sign: \[ (4x + 1)^{3/2} = -27 \]

Step 2: Raise Both Sides to the Power of \(2/3\)

Raise both sides to the power of \(\frac{2}{3}\) to eliminate the exponent: \[ 4x + 1 = (-27)^{2/3} \]

Step 3: Simplify the Right-Hand Side

Calculate \((-27)^{2/3}\): \[ (-27)^{2/3} = 9 \] Thus, the equation becomes: \[ 4x + 1 = 9 \]

Step 4: Solve for \(x\)

Isolate \(x\) by subtracting 1 from both sides and then dividing by 4: \[ 4x = 8 \implies x = 2 \]

Final Answer

\[ \boxed{x = 2} \]

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