Question

How do you write $ {{y}^{\dfrac{5}{4}}} $ in radical form?

Answer 1

Hint:
 In this question, we need to write $ {{y}^{\dfrac{5}{4}}} $ in radical form. For this, we need to write it in the form of underroot sign $ \left( \sqrt[x]{y} \right) $ . We will first use the law of exponents i.e. $ {{\left( {{x}^{a}} \right)}^{b}}={{x}^{ab}} $ and separate $ \dfrac{1}{4},5 $ . After that, we will convert power $ \dfrac{1}{4} $ to radical form keeping $ {{y}^{5}} $ inside it. We will use property that $ \sqrt[x]{y}={{y}^{\dfrac{1}{x}}} $ . This will give us the required form.

Complete step by step answer:
Here we are given the expression as $ {{y}^{\dfrac{5}{4}}} $ . We need to convert it into radical form.
Let us first understand the meaning of radical form. Radical form refers to a form of an algebraic expression in which we have a number or an expression underneath a radical (under root sign). Now we have the expression as $ {{y}^{\dfrac{5}{4}}} $ .
We can write it as $ {{y}^{5\cdot \dfrac{1}{4}}} $ . We know that from the law of exponents that, $ {{\left( {{x}^{a}} \right)}^{b}}={{x}^{ab}} $ so applying the reverse of this we get $ {{y}^{5\cdot \dfrac{1}{4}}}={{\left( {{y}^{5}} \right)}^{\dfrac{1}{4}}} $ .
According to the rule of exponents and radicals to write the expression in radical form we know that $ \sqrt[x]{y}={{y}^{\dfrac{1}{x}}} $ . So we can apply it for our expression $ {{\left( {{y}^{5}} \right)}^{\dfrac{1}{4}}} $ . We get $ \sqrt[4]{{{y}^{5}}} $ .
As we can see the term is under the radical (underroot sign). So our required radical form is $ {{y}^{\dfrac{5}{4}}}=\sqrt[4]{{{y}^{5}}} $ .

Note:
Students can also simplify the found radical form in the following way,
As we know $ {{x}^{a}}\cdot {{x}^{b}}={{x}^{a+b}} $ so we can write $ {{y}^{5}}={{y}^{1+4}}=y\cdot {{y}^{4}} $ . Therefore, $ \sqrt[4]{{{y}^{5}}}=\sqrt[4]{{{y}^{4}}\cdot y} $ .
Using the rule of radical $ \sqrt[x]{a\cdot b}=\sqrt[x]{a}\cdot \sqrt[x]{b} $ we get $ \sqrt[4]{{{y}^{5}}}=\sqrt[4]{{{y}^{4}}}\cdot \sqrt[4]{y} $ .
 $ \sqrt[4]{{{y}^{4}}} $ can be written as \[{{\left( {{y}^{4}} \right)}^{\dfrac{1}{4}}}={{y}^{\dfrac{4}{4}}}={{y}^{1}}=y\]. So our expression becomes $ \sqrt[4]{{{y}^{5}}}=y\cdot \sqrt[4]{y} $ .
So $ y\cdot \sqrt[4]{y} $ is the simplified radical form. Students should keep in mind the various laws of exponents and rule of radical to solve this sum.