Question

Express 15 x^1/3y^1/5 using a radical

Answer 1

in short, to convert two fractions to have the same denominator, we simply multiply one by the denominator of the other, so in this case, we'll multiply 1/3 by 5, top and bottom, and 1/5 by 3, top and bottom, thus


`\bf a^{( n)/( m)} \implies \sqrt[ m]{a^ n} \qquad \qquad \sqrt[ m]{a^ n}\implies a^{( n)/( m)}\\\\ -------------------------------`


`\bf \cfrac{15x^{(1)/(3)}}{y^{(1)/(5)}}\qquad \begin{cases} (1)/(3)=(1\cdot 5)/(3\cdot 5)\\ \qquad (5)/(15)\\\\ (1)/(5)=(1\cdot 3)/(5\cdot 3)\\ \qquad (3)/(15) \end{cases}\implies \cfrac{15x^{(5)/(15)}}{y^{(3)/(15)}}\implies 15\cdot \cfrac{x^{(5)/(15)}}{y^{(3)/(15)}}\implies 15\cdot \cfrac{\sqrt[15]{x^5}}{\sqrt[15]{y^3}} \\\\\\ 15\sqrt[15]{(x^5)/(y^3)}`

Answer 2

`15\cdot\sqrt[3]{x}\cdot \sqrt[5]{y}`

Explanation:

We have been given an expression
`15x^{(1)/(3)}y^{(1)/(5)`. We are asked to express our given expression using a radical.

Using fractional exponent rule
`a^{(m)/(n)}=\sqrt[n]{a^m}`, we can write terms of our given expression as:

`15\cdot\sqrt[3]{x^1}\cdot \sqrt[5]{y^1}`

`15\cdot\sqrt[3]{x}\cdot \sqrt[5]{y}`

Therefore, our required expression would be
`15\cdot\sqrt[3]{x}\cdot \sqrt[5]{y}`.