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Question: Simplify the following surd: \({\left( {\sqrt 5 } \right)^2}\)...

Simplify the following surd: (5)2{\left( {\sqrt 5 } \right)^2}

Explanation

Solution

We have to simply solve the surd and the power which is placed over the number. After simplifying the surd (square root) we get an index of 12\dfrac{1}{2}and then we multiply it with the index placed over the bracket of the given expression.

Complete solution step by step:
Firstly, we write the given expression
(5)2{\left( {\sqrt 5 } \right)^2}
Simplifying the bracket part first using the square root simplification i.e. when we remove the square root, we replace it by index (power) of 12\dfrac{1}{2} in this manner
p=(p)12 ⇒((5)12)2  \sqrt p = {(p)^{\dfrac{1}{2}}} \\\ \Rightarrow {\left( {{{(5)}^{\dfrac{1}{2}}}} \right)^2} \\\
Now to solve this expression we use the index properties i.e.
(pq)r=(pr)q=pq×r{({p^q})^r} = {({p^r})^q} = {p^{q \times r}}
Using this property and taking first two parts of it we solve our expression
(512)2=(52)12 =(25)12  {\left( {{5^{\dfrac{1}{2}}}} \right)^2} = {\left( {{5^2}} \right)^{\dfrac{1}{2}}} \\\ = {(25)^{\dfrac{1}{2}}} \\\
Converting the index into square root we have
(25)12=25 =5  {(25)^{\dfrac{1}{2}}} = \sqrt {25} \\\ = 5 \\\
Our answer comes out to be ‘5’. Now we check it by taking first and third part of the property
(512)2=(5)12×2 =5  {\left( {{5^{\dfrac{1}{2}}}} \right)^2} = {(5)^{\dfrac{1}{2} \times 2}} \\\ = 5 \\\
Our answer comes out the same as the previous result i.e. ‘5’.

Additional information: We can solve the problem using basic index method also in which we solve the index part of the expression as explained below
(5)2{\left( {\sqrt 5 } \right)^2}
The expression has an index of 2 which means the number inside the bracket will be multiplied twice with itself so we have -
(5)2=5×5{\left( {\sqrt 5 } \right)^2} = \sqrt 5 \times \sqrt 5
Now we use multiplication property of square roots i.e.
p×p=p\sqrt p \times \sqrt p = p
Now using the property we have
(5)2=5×5=5{\left( {\sqrt 5 } \right)^2} = \sqrt 5 \times \sqrt 5 = 5
Our answer comes out to be ‘5’, the same as when calculated by the previous method.

Note: In laymen terms index is the power raised to a number and surd is the ‘nth’ root of a number i.e. an=(a)1n\sqrt[n]{a} = {(a)^{\dfrac{1}{n}}} and we can say both are opposite in nature because after solving index (power) of a number will be the raised value of the number whereas surd will become the reduced root value of the number.