Question

simplify the radical. assume that all variables represent positive real numbers.
$sqrt3{-27x^{9}y^{6}}$
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. $sqrt3{-27x^{9}y^{6}} = square$
b. the radical does not represent a real number.

Explanation:

Step1: Analyze the cube root

We know that for a cube root \(\sqrt[3]{abc}=\sqrt[3]{a}\cdot\sqrt[3]{b}\cdot\sqrt[3]{c}\). So we can split \(\sqrt[3]{-27x^{9}y^{6}}\) into \(\sqrt[3]{-27}\cdot\sqrt[3]{x^{9}}\cdot\sqrt[3]{y^{6}}\).

Step2: Simplify each cube root

• For \(\sqrt[3]{-27}\), since \((-3)^3 = -27\), so \(\sqrt[3]{-27}=-3\).
• For \(\sqrt[3]{x^{9}}\), using the property \(\sqrt[n]{a^{m}}=a^{\frac{m}{n}}\), here \(n = 3\) and \(m=9\), so \(\sqrt[3]{x^{9}}=x^{\frac{9}{3}}=x^{3}\).
• For \(\sqrt[3]{y^{6}}\), using the same property, \(n = 3\) and \(m = 6\), so \(\sqrt[3]{y^{6}}=y^{\frac{6}{3}}=y^{2}\).

Step3: Multiply the simplified terms

Multiply \(-3\), \(x^{3}\) and \(y^{2}\) together, we get \(-3x^{3}y^{2}\). And since we can find a real number (in this case \(-3x^{3}y^{2}\)) to represent the cube root, the radical does represent a real number, so option A is correct and we fill in \(-3x^{3}y^{2}\).

Answer:

A. \(\boldsymbol{-3x^{3}y^{2}}\)