Questions: Use the properties of radicals to simplify each expression. (a) sqrt(48) * sqrt(3) (b) sqrt[6]((7 x^2)^6)

Solution Steps

To simplify the given radical expressions, we can use the properties of radicals. Specifically, we can use the property that the product of two square roots is the square root of the product, and the property that the nth root of a number raised to the nth power is the number itself.

(a) For \(\sqrt{48} \cdot \sqrt{3}\), we can combine the radicals under a single square root.

(b) For \(\sqrt[6]{\left(7 x^{2}\right)^{6}}\), we can use the property that the sixth root of something raised to the sixth power simplifies to the base itself.

Using the property of radicals that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\), we can combine the radicals: \[ \sqrt{48} \cdot \sqrt{3} = \sqrt{48 \cdot 3} = \sqrt{144} \] Since \(\sqrt{144} = 12\), we have: \[ \sqrt{48} \cdot \sqrt{3} = 12.0 \]

Using the property that \(\sqrt[n]{a^n} = a\), we can simplify: \[ \sqrt[6]{\left(7 x^{2}\right)^{6}} = 7 x^{2} \] Thus, the simplified form is: \[ \sqrt[6]{\left(7 x^{2}\right)^{6}} = 28 \]

Final Answer