To solve the equation \(\frac{1}{5} x^{3/4} - 31 = -6\), we can follow these steps:
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Add 31 to both sides: \[ \frac{1}{5} x^{3/4} - 31 + 31 = -6 + 31 \] \[ \frac{1}{5} x^{3/4} = 25 \]
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Multiply both sides by 5: \[ 5 \cdot \frac{1}{5} x^{3/4} = 25 \cdot 5 \] \[ x^{3/4} = 125 \]
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Raise both sides to the power of \( \frac{4}{3} \) to solve for \( x \): \[ \left(x^{3/4}\right)^{\frac{4}{3}} = 125^{\frac{4}{3}} \] \[ x = 125^{\frac{4}{3}} \]
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Calculate \( 125^{\frac{4}{3}} \): Note that \( 125 = 5^3 \), so: \[ 125^{\frac{4}{3}} = (5^3)^{\frac{4}{3}} = 5^{3 \cdot \frac{4}{3}} = 5^4 = 625 \]
Thus, the solution to the equation is: \[ \boxed{625} \]
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