Questions: Simplify the following expression. Assume that each variable is positive. (4-16 sqrt(54 y^3))/16

Solution Steps

The given expression is: \[ \frac{4 - 16 \sqrt{54 y^{3}}}{16} \] First, factor out the common factor in the numerator. Both terms in the numerator have a factor of 4: \[ 4 - 16 \sqrt{54 y^{3}} = 4(1 - 4 \sqrt{54 y^{3}}) \]

The denominator is 16. Divide both terms in the numerator by 16: \[ \frac{4(1 - 4 \sqrt{54 y^{3}})}{16} = \frac{1 - 4 \sqrt{54 y^{3}}}{4} \]

Simplify \( \sqrt{54 y^{3}} \): \[ \sqrt{54 y^{3}} = \sqrt{54} \cdot \sqrt{y^{3}} = \sqrt{9 \cdot 6} \cdot y^{3/2} = 3 \sqrt{6} \cdot y^{3/2} \] Substitute this back into the expression: \[ \frac{1 - 4 \cdot 3 \sqrt{6} \cdot y^{3/2}}{4} = \frac{1 - 12 \sqrt{6} \cdot y^{3/2}}{4} \]

Final Answer