Question: Simplify. \(\sqrt{48x^7y^{10}}\) Assume that all variables represent positive real numbers.

Simplify. \(\sqrt{48x^7y^{10}}\) Assume that all variables represent positive real numbers.

Solution

To simplify the expression \(\sqrt[3]{48x^7y^{10}}\), follow these steps: First, factor the numbers and the variables to ensure you are taking the cube root of perfect cubes where possible. 48 can be factored as: \[ 48 = 2^4 \times 3^1 \] For the variables: \[ x^7 = x^6 \times x^1 = (x^2)^3 \times x^1 \] \[ y^{10} = y^9 \times y^1 = (y^3)^3 \times y^1 \] Now the expression is: \[ \sqrt[3]{48x^7y^{10}} = \sqrt[3]{2^4 \times 3^1 \times (x^2)^3 \times x^1 \times (y^3)^3 \times y^1} \] We can extract the cube roots: \[ = \sqrt[3]{2^3 \times 2^1 \times 3^1 \times (x^2)^3 \times x^1 \times (y^3)^3 \times y^1} \] \[ = 2 \times x^2 \times y^3 \times \sqrt[3]{2 \times 3 \times x \times y} \] So, the simplified form is: \[ 2x^2y^3 \sqrt[3]{6xy} \]