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In $3-38$ , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero.$$\sqrt[4]{\frac{2 a^{8}}{b^{2} c^{3}}}$$

$\frac{a^{2} \sqrt[4]{2 b^{2} c}}{b c}$

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Chapter 3

REAL NUMBERS AND RADICALS

Section 3

Simplifying Radicals

Whole which of Numbers

Fractions and Mixed Numbers

Decimals

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In $3-38$ , write each rad…

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if we have the expression the fourth root of to A to the eighth over B squared C to the third. We want to simplify this well, according to some properties of radicals A. When we're rooting a fraction, we can take the in this case, fourth root of the numerator over the fourth root of the denominator. No, we can take the fourth root of aid to the eighth. That is a squared, so we have a square times before through of two over the fourth root of B squared C Cube. Now we need to rationalize the denominator, so we need to scale both the numerator and denominator by a radical that's going to remove it from the denominator so we can only multiply four through its by fourth roots. And if if fourth roots are going to a value, excuse me, evaluate powers of four that we have to think about, what could we multiply? What's underneath the radical to get powers of four. So if you already have a B squared, if we do, another B squared will have be to the fourth, which then we can evaluate the B and if we already have C to the third. We just need to multiply by another C to get seated the fourth. Therefore we could multiply it. So that's what we're going to scale in order to rationalize the denominator. Yeah, And so now what we have here in the numerator, we still have that a squared. And then we can multiply fourth routes together, we'll have times the fourth root of two B squared C over. Now, this is the fourth through a B to the fourth, which is B and the fourth root of sea to the fourth, which is C.

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