Question

Find the square root. Assume that the variable is unrestricted, and use absolute value symbols when necessary. (Simplify your answer completely)

Answer 1

We are given the following expression:

$\sqrt[]{81x^2}$

To simplify this expression we will use the following property of radicals:

$\sqrt[]{ab}=\sqrt[]{a}\sqrt[]{b}$

Applying the property we get:

$\sqrt[]{81x^2}=\sqrt[]{81}\sqrt[]{x^2}$

Now, the first radical is equal to 9 since 9 x 9 = 81, therefore, we get:

$\sqrt[]{81x^2}=\sqrt[]{81}\sqrt[]{x^2}=9\sqrt[]{x^2}$

For the second radical we will use the following property of absolute values:

$\lvert x\rvert=\sqrt[]{x^2}$

Replacing we get:

$\sqrt[]{81x^2}=\sqrt[]{81}\sqrt[]{x^2}=9\sqrt[]{x^2}=9\lvert x\rvert$

Therefore, the expression reduces to the product of 9 and the absolute value of "x".