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(sqrt [3](32m^6))(sqrt [3](3125m^7))

Pergunta

(sqrt [3](32m^6))(sqrt [3](3125m^7))

(sqrt [3](32m^6))(sqrt [3](3125m^7))

Solução

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GabrielMestre · Tutor por 5 anos

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To simplify the expression (\sqrt[3]{32m^6})(\sqrt[3]{3125m^7}), we can use the property of cube roots that states \sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}.

First, let's multiply the expressions inside the cube roots:

\sqrt[3]{32m^6} \cdot \sqrt[3]{3125m^7} = \sqrt[3]{32 \cdot 3125 \cdot m^6 \cdot m^7}

Next, we simplify the product inside the cube root:

32 \cdot 3125 = 100000

m^6 \cdot m^7 = m^{6+7} = m^{13}

So, the expression becomes:

\sqrt[3]{100000 \cdot m^{13}}

Now, we can simplify the cube root:

\sqrt[3]{100000 \cdot m^{13}} = \sqrt[3]{100000} \cdot \sqrt[3]{m^{13}}

We know that:

\sqrt[3]{100000} = 100 \quad \text{and} \quad \sqrt[3]{m^{13}} = m^{13/3} = m^{4 + 1/3} = m^4 \cdot m^{1/3}

Therefore, the simplified expression is:

100 \cdot m^4 \cdot \sqrt[3]{m}

So, the final answer is:

100m^4 \sqrt[3]{m}
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