Question

Can you explain to me how to solve this please

Answer 1

To solve the exercise you can use the following properties of exponents and radicals

`a^{(m)/(n)}=\sqrt[n]{a^m}`
`a^ma^n=a^(m+n)`
`(a^m)/(a^n)=a^(m-n)`

So, you have

`\begin{gathered} \frac{\sqrt[3]{7}\sqrt[]{7}}{\sqrt[6]{7^5}}=\frac{7^{(1)/(3)}\cdot7^{(1)/(2)}}{\sqrt[6]{7^5}} \\ \frac{\sqrt[3]{7}\sqrt[]{7}}{\sqrt[6]{7^5}}=\frac{7^{(1)/(3)+(1)/(2)}}{7^{(5)/(6)}} \\ \frac{\sqrt[3]{7}\sqrt[]{7}}{\sqrt[6]{7^5}}=\frac{7^{(5)/(6)}}{7^{(5)/(6)}} \\ \frac{\sqrt[3]{7}\sqrt[]{7}}{\sqrt[6]{7^5}}=7^{(5)/(6)-(5)/(6)} \\ \frac{\sqrt[3]{7}\sqrt[]{7}}{\sqrt[6]{7^5}}=7^0 \end{gathered}`

By definition

`\begin{gathered} a^0=1 \\ \text{Where a }\\e0 \end{gathered}`

So

`\begin{gathered} \frac{\sqrt[3]{7}\sqrt[]{7}}{\sqrt[6]{7^5}}=7^0 \\ \frac{\sqrt[3]{7}\sqrt[]{7}}{\sqrt[6]{7^5}}=1 \end{gathered}`

Therefore, the correct answer is b. 1.