Pergunta
)(sqrt [3](3125m^7))](https://static.questionai.br.com/resource%2Fqaiseoimg%2F202501%2Fsqrt-332m6sqrt-33125m7-t2ZaSgkUSQ0h.jpg?x-oss-process=image/resize,w_558,h_500/quality,q_35/format,webp)
(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}
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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