
How do you simplify the square root of Y to the 5th power?
Answer
445.8k+ views
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.
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.
Recently Updated Pages
What percentage of the area in India is covered by class 10 social science CBSE

The area of a 6m wide road outside a garden in all class 10 maths CBSE

What is the electric flux through a cube of side 1 class 10 physics CBSE

If one root of x2 x k 0 maybe the square of the other class 10 maths CBSE

The radius and height of a cylinder are in the ratio class 10 maths CBSE

An almirah is sold for 5400 Rs after allowing a discount class 10 maths CBSE

Trending doubts
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Why is there a time difference of about 5 hours between class 10 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

What constitutes the central nervous system How are class 10 biology CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Explain the Treaty of Vienna of 1815 class 10 social science CBSE
