Questions: Question 12 Simplify. Assume all variables are nonnegative. √(48 b^12 y^5 / b^5 y^15)=

Solution Steps

To simplify the given expression, we will first simplify the fraction inside the square root by dividing the exponents of like bases. Then, we will take the square root of the resulting expression by applying the square root to both the coefficient and the variables separately.

We start with the expression

\[ \sqrt{\frac{48 b^{12} y^{5}}{b^{5} y^{15}}} \]

First, we simplify the fraction inside the square root by dividing the coefficients and the variables separately.

The coefficient \(48\) can be factored as \(16 \cdot 3\), and since \(16\) is a perfect square, we can take its square root:

\[ \sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3} \]

Next, we simplify the variables:

\[ \frac{b^{12}}{b^{5}} = b^{12-5} = b^{7} \]

\[ \frac{y^{5}}{y^{15}} = y^{5-15} = y^{-10} = \frac{1}{y^{10}} \]

Combining these results, we have:

\[ \sqrt{\frac{48 b^{12} y^{5}}{b^{5} y^{15}}} = \sqrt{4\sqrt{3} \cdot b^{7} \cdot \frac{1}{y^{10}}} = 4\sqrt{3} \cdot \frac{b^{7/2}}{y^{5}} \]

Final Answer