4. Simplifique as potencias a) $$ \left(125^{\frac{2}{3}}+16^{\frac{1}{2}}+343^{\frac{4}{3}}\right)^{\frac{1}{2}} $$

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1. Observe que $$125 = 5^3$$. Assim, temos:
$$
125^{\frac{2}{3}} = \left(5^3\right)^{\frac{2}{3}} = 5^2 = 25
$$

2. Para $$16^{\frac{1}{2}}$$, temos:
$$
16^{\frac{1}{2}} = \sqrt{16} = 4
$$

3. Note que $$343 = 7^3$$. Portanto:
$$
343^{\frac{4}{3}} = \left(7^3\right)^{\frac{4}{3}} = 7^4 = 2401
$$

4. Somando os termos:
$$
125^{\frac{2}{3}} + 16^{\frac{1}{2}} + 343^{\frac{4}{3}} = 25 + 4 + 2401 = 2430
$$

5. A expressão original se torna:
$$
\left(2430\right)^{\frac{1}{2}} = \sqrt{2430}
$$

6. Para simplificar $$\sqrt{2430}$$, fatoramos 2430:
$$
2430 = 3^5 \times 2 \times 5 = 3^4 \times (3 \times 2 \times 5) = 81 \times 30
$$
$$
\sqrt{2430} = \sqrt{81 \times 30} = \sqrt{81} \times \sqrt{30} = 9\sqrt{30}
$$

Portanto, a expressão simplificada é:
$$
9\sqrt{30}
$$
A expressão simplificada é $$9\sqrt{30}$$.

4. Simplifique as potencias a) \( \left(125^{\frac{2}{3}}+16^{\frac{1}{2}}+343^{\frac{4}{3}}\right)^{\frac{1}{2}} \)

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