Question

select the expression that is equivalent to \\(\sqrt3{10^4} \cdot \sqrt3{10}\\).
answer
\\(100^{\frac{5}{3}}\\) \\(10^{\frac{4}{9}}\\)
\\(100^{\frac{4}{9}}\\) \\(10^{\frac{5}{3}}\\)

Explanation:

Step1: Recall the property of radicals and exponents

The cube root of a number \(a\) can be written as \(a^{\frac{1}{3}}\), so \(\sqrt[3]{10^{4}} = 10^{\frac{4}{3}}\) and \(\sqrt[3]{10}=10^{\frac{1}{3}}\).

Step2: Use the property of exponents \(a^{m}\cdot a^{n}=a^{m + n}\)

When multiplying two exponential expressions with the same base, we add the exponents. So \(10^{\frac{4}{3}}\cdot10^{\frac{1}{3}}=10^{\frac{4}{3}+\frac{1}{3}}\).

Step3: Simplify the exponent

\(\frac{4}{3}+\frac{1}{3}=\frac{4 + 1}{3}=\frac{5}{3}\), so the expression simplifies to \(10^{\frac{5}{3}}\).

Answer:

\(10^{\frac{5}{3}}\) (the last option)