Question

select the expression that is equivalent to (sqrt6{10} cdot sqrt6{10^5}).

answer

(\bigcirc) 10 (\bigcirc) 100
(\bigcirc) (10^{\frac{5}{36}}) (\bigcirc) (100^{\frac{5}{36}})

Explanation:

Step1: Recall the property of radicals

The property of radicals states that \(\sqrt[n]{a}\cdot\sqrt[n]{b}=\sqrt[n]{ab}\) and also \(\sqrt[n]{a}=a^{\frac{1}{n}}\). So we can rewrite the given expression \(\sqrt[6]{10}\cdot\sqrt[6]{10^{5}}\) using the first property as \(\sqrt[6]{10\times10^{5}}\).

Step2: Simplify the exponent inside the radical

Using the rule of exponents \(a^{m}\cdot a^{n}=a^{m + n}\), for \(10\times10^{5}\), we have \(10^{1+5}=10^{6}\). So the expression becomes \(\sqrt[6]{10^{6}}\).

Step3: Simplify the sixth root of \(10^{6}\)

Using the property \(\sqrt[n]{a^{n}}=a\) (when \(n\) is a positive integer and \(a\) is a real number), for \(\sqrt[6]{10^{6}}\), we get \(10\).

Answer:

A. 10