Question

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers.5∛2−2∛16+1∛54\frac{5}{∛2}-\frac{2}{∛16}+\frac{1}{∛54}

Accepted Answer

Rewrite each cube root in the denominators as an exponent of 1/3. For example, express $$\sqrt[3]{2} as$$ $$2^{1/3}$$, $$\sqrt[3]{16} as$$ $$16^{1/3}$$, and $$\sqrt[3]{54} as$$ $$54^{1/3}. $$This gives the expression: $$\frac{5}{2$$^{1/3}$$} - \frac{2}{16$$^{1/3}$$} + \frac{1}{54$$^{1/3}$$}. $$Simplify the radicands where possible by expressing them as products of prime factors or perfect cubes. For example, $$16 = 2^4$ and 54 = 2$$ 	imes $$3^3. $$Use this to rewrite the cube roots: $$16^{1/3}$$ = $$(2^4$$)^{1/3}$$ = 2$$^{4/3}$$ and 54$$^{1/3}$$ = (2 	imes 3^3$)^{1/3}$$ = $$2^{1/3}$$ 	imes 3$$. Rewrite each term using the simplified exponents: $$\frac{5}{$$2^{1/3}$$} - \frac{2}{$$2^{4/3}$$} + \frac{1}{$$2^{1/3}$$ 	imes 3}$$. This will help in combining like terms later. Find a common denominator for all three terms. Since the denominators involve powers of 2 and possibly 3, the common denominator will be the least common multiple of 2$$^{1/3}$$, 2$$^{4/3}$$, and 2$$^{1/3}$$ 	imes 3. $$Express each term with this common denominator. Combine the numerators over the common denominator by performing the indicated addition and subtraction. After combining, simplify the numerator if possible, and write the final expression as a single fraction.

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers.
$\frac{5}{∛ 2} - \frac{2}{∛ 16} + \frac{1}{∛ 54}$

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Properties of Radicals and Cube Roots

Operations with Fractions

Simplification of Expressions with Variables and Radicals

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers.5∛2−2∛16+1∛54\(\frac{5}{∛2}\)-\(\frac{2}{∛16}\)+\(\frac{1}{∛54}\)