Simplify the radical expressions in Exercises 58 - 62. ∜(32x5)/∜(16x) (Assume that x \u003e 0.)

Rewrite the expression by combining the fourth roots into a single fourth root: $$\frac{\sqrt[4]{32x^5$}}{\sqrt[4]{16x}} = \sqrt[4]{\frac{32x^5$}{16x}}. $$Simplify the fraction inside the radical: $$\frac{32x^5$}{16x} = 2x$$^{5-1}$$ = 2x^4$. Now the expression is $$\sqrt[4]{$$2x^4$$}$$. Recognize that $$\sqrt[4]{$$x^4$$} = x$$ since x$$ \u003e 0$$. Separate the fourth root into the product of two fourth roots: $$\sqrt[4]{$$2x^4$$} = \sqrt[4]{2} \times \sqrt[4]{$$x^4$$} = \sqrt[4]{2} \times x$$. Write the simplified expression as x$$ \sqrt[4]{2}$$, which is the simplified form of the original expression.

Ch. P - Fundamental Concepts of Algebra

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Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

Problema 62

Capitolo 1, Problema 62

Simplify the radical expressions in Exercises 58 - 62. ∜(32x⁵)/∜(16x) (Assume that x > 0.)

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Rewrite the expression by combining the fourth roots into a single fourth root: $\frac{\sqrt[4]{32x^5}}{\sqrt[4]{16x}} = \sqrt[4]{\frac{32x^5}{16x}}$.

Simplify the fraction inside the radical: $\frac{32x^5}{16x} = 2x^{5-1} = 2x^4$.

Now the expression is $\sqrt[4]{2x^4}$. Recognize that $\sqrt[4]{x^4} = x$ since $x > 0$.

Separate the fourth root into the product of two fourth roots: $\sqrt[4]{2x^4} = \sqrt[4]{2} \times \sqrt[4]{x^4} = \sqrt[4]{2} \times x$.

Write the simplified expression as $x \sqrt[4]{2}$, which is the simplified form of the original expression.

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Properties of Radicals

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When simplifying radicals containing variables, apply the root to both the coefficient and the variable separately. For variables with exponents, divide the exponent by the root's index, extracting whole powers outside the radical and leaving the remainder inside.

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The fourth root (⁴√) of a number is the value that, when raised to the fourth power, returns the original number. Using exponent rules, ⁿ√(a^m) = a^(m/n), helps convert radicals into fractional exponents, simplifying calculations and expressions.

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