Question

How do you simplify the square root of Y to the 5th power?

Answer 1

Hint: Here in this question. First we write the given variable or number y in the form of radical expression. Then to simplify, expand the radical expression using the product rule of exponent function i.e., \[{x^{a + b}} = {x^a}{x^b}\] and on further simplification we get the required solution.

Complete step-by-step solution:
A radical expression is a numerical expression or an algebraic expression that includes a radical. a radical expression and its index an index of 5 means that we are looking for the fifth root. An index of 3 means that we are looking for the cube root. An index of 2 is the square root.
Consider the given question i.e., square root of Y to the 5th power can be written as:
\[ \Rightarrow \,\,\,\sqrt {{y^5}} \] (1)
By using the product rule of exponent function i.e., if \[x \ne 0\] then \[{x^{a + b}} = {x^a}{x^b}\].
Then \[{y^5}\] can be written as
\[ \Rightarrow \,\,{y^5} = {y^{2 + 2 + 1}}\]
\[ \Rightarrow \,\,{y^5} = {y^2} \cdot {y^2} \cdot {y^1}\]
Hence equation (1) can be written as
\[ \Rightarrow \,\,\,\sqrt {{y^5}} = \sqrt {{y^2} \cdot {y^2} \cdot {y^1}} \] (2)
By using Multiplication property of square root i.e., \[\sqrt {ab} = \sqrt a .\sqrt b \]
Equation (2) can be written as
\[ \Rightarrow \,\,\,\sqrt {{y^2}} \cdot \sqrt {{y^2}} .\sqrt y \]
\[ \Rightarrow \,\,\,{\left( {\sqrt {{y^2}} } \right)^2}.\sqrt y \] (3)
By the power rule of exponent i.e., \[\,{\left( {\sqrt a } \right)^2} = a\]
Equation (3) can be written as
\[ \Rightarrow \,\,\,{y^2}.\sqrt y \]
Hence, the simplified form of square root of Y to the 5th power or \[\sqrt {{y^5}} \] is \[{y^2}.\sqrt y \].

Note: The number is having a power n, then it is written n times a number. there is a law of indices on the exponents. The law of exponents is applied to the exponential number also. The square root and the square are both inverse to each other and hence the square root and square will get canceled.