What is the following product? Assume x ≥ 0 \sqrt[3]{x^{2} } . \sqrt[4]{x^{3} } A. x\sqrt{x} B.

Question

What is the following product? Assume x ≥ 0

\sqrt[3]{x^{2} } . \sqrt[4]{x^{3} }

A. x\sqrt{x}

B. \sqrt[12]{x^{5} }

C. x(\sqrt[12]{x^{5} } )

D. x6

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Sarah 1 week 2021-10-10T20:50:19+00:00 1 Answer 0

Answers ( )

    0
    2021-10-10T20:51:57+00:00

    For this case we must multiply the following expression:

    \sqrt [3] {x ^ 2} * \sqrt [4] {x ^ 3}

    By definition of properties of otencias and roots we have:

    \sqrt [n] {a ^ m} = a ^ {\frac {m} {n}}

    We rewrite the terms of the expression:

    \sqrt [3] {x ^ 2} = (x ^ 2) ^ {\frac {1} {3}} = (x ^ 2) ^ {\frac {4} {12}}\\\sqrt [4] {x ^ 3} = (x ^ 3) ^ {\frac {1} {4}} = (x ^ 3) ^ {\frac {3} {12}}

    So, we have:

    (x ^ 2) ^ {\frac {4} {12}} * (x ^ 3) ^ {\frac {3} {12}} =

    Applying the above definition we have:

    \sqrt [12] {(x ^ 2) ^ 4} * \sqrt [12] {(x ^ 3) ^ 3} =

    We multiply the exponents:

    \sqrt [12] {x ^ 8} * \sqrt [12] {x ^ 9} =

    We combine using the product rule for radicals.

    \sqrt [12] {x ^ 8 * x ^ 9} =

    By definition of multiplication properties of powers of the same base, we put the same base and add the exponents:

    \sqrt [12] {x ^ {8 + 9}} =\\\sqrt [12] {x ^ {17}} =\\\sqrt [12] {x ^ {12} * x ^ 5} =\\x \sqrt [12] {x ^ 5}

    Answer:

    Option C

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