Question

Why can’t the square root of ${a^2} + {b^2}$ be simplified?

Answer 1

Hint: In the given question, we need to prove that the square root of ${a^2} + {b^2}$ can’t be simplified. It can also be written as $\sqrt {{a^2} + {b^2}} $. We will try to prove this using an example. We will substitute any number as $a$ and $b$ and simplify it to prove that the square root of ${a^2} + {b^2}$ can’t be simplified.

Complete step by step answer:
Suppose if we substitute $a$ and $b$ equal to $5$, then we get
$ \Rightarrow \sqrt {{a^2} + {b^2}} = \sqrt {{5^2} + {5^2}} $
On solving the squaring terms inside the root, we get
$ \Rightarrow \sqrt {{a^2} + {b^2}} = \sqrt {25 + 25} $
In addition of terms inside the square root, we get
$ \Rightarrow \sqrt {{a^2} + {b^2}} = \sqrt {50} $
$\sqrt {50} $ is an irrational number.
However if it was $\sqrt {{5^2}} + \sqrt {{5^2}} $ it would be equal to $5 + 5 = 25$, as $\sqrt {} $ and $^2$ would cancel out to give the equation $5 + 5$.
So if putting $a$ and $b$ into our simpler expression only involve addition, subtraction, multiplication and/or division of terms with rational coefficients then we would not be able to produce $\sqrt 5 $.
Therefore, any expression for $\sqrt {{a^2} + {b^2}} $ must involve something beyond addition, subtraction, multiplication and/or division of terms with rational coefficients.
Therefore, $\sqrt {{a^2} + {b^2}} $ cannot be simplified unless given a substitution for $a$ and $b$.

Note:
Remember that if we add $2ab$ inside $\sqrt {{a^2} + {b^2}} $, we can write it as $\sqrt {{a^2} + {b^2} + 2ab} $. As we know ${a^2} + {b^2} + 2ab = {\left( {a + b} \right)^2}$, therefore we get $\sqrt {{{\left( {a + b} \right)}^2}} $ which is equal to $\left( {a + b} \right)$. Similarly, if we subtract $2ab$ inside $\sqrt {{a^2} + {b^2}} $, we can write it as $\sqrt {{a^2} + {b^2} - 2ab} $. As we know ${a^2} + {b^2} - 2ab = {\left( {a - b} \right)^2}$, therefore we get $\sqrt {{{\left( {a - b} \right)}^2}} $ which is equal to $\left( {a - b} \right)$. This proves that the square root of ${a^2} + {b^2}$ can only be simplified if substitute terms in form of $a$ and $b$.