We need to calculate: the simplified form of (3sqrt[3]{a^{3}}(-5sqrt[4]{a^{3}}))

Question

We need to calculate: the simplified form of

(3 \sqrt[3]{a^{3}} (- 5 \sqrt[4]{a^{3}}))

Answer 1

Given:

(3 \sqrt[3]{a^{3}} (- 5 \sqrt[4]{a^{3}}))

Used formula:
For a real number a when m is even,

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

Calculation:

(3 \sqrt[3]{a^{3}} (- 5 \sqrt[4]{a^{3}}))

The radicals are like radicals and have the same radicand &same index. Reprouping the factors and applying the product property of radicals:

(3 \sqrt[3]{a^{3}} (- 5 \sqrt[4]{a^{3}})) = 3 \times (- 5) \sqrt[4]{a^{3} \cdot a^{3}}

Apply property of exponents:

3 \times (- 5) \sqrt[4]{a^{3} \cdot a^{3}} - 15 \sqrt[4]{a^{6}}

- 15 \sqrt[4]{a^{6}} = 15 \sqrt[4]{a^{4} \cdot a^{2}}

= - 15 a \sqrt[4]{a^{2}}

Rewrite the expression with rational exponent:

= - 15 a \sqrt[4]{a^{2}} = - 15 a (a^{2})^{\frac{1}{4}}

= - 15 a {\sqrt[4]{a^{2}}}^{\frac{1}{4}} = - 15 a \cdot a^{2 \times \frac{1}{4}}

= - 15 a \cdot a^{\frac{1}{2}}

= - 15 a \cdot a^{\frac{1}{2}} = - 15 a \cdot \sqrt{a}

= - 15 a \sqrt{a}

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