Question

How do you simplify \[\sqrt{25-{{x}^{2}}}\]

Answer 1

Hint: Write ‘25’ as the square of ‘5’ i.e. ${{\left( 5 \right)}^{2}}$. Then apply the formula ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a+b \right)$. Do the necessary simplification by separating the radical using the formula $\sqrt{a\cdot b}=\sqrt{a}\cdot \sqrt{b}$, so that both factors should be in multiplication form with each other. Obtain the required solution from there.

Complete step by step answer:
The expression we have, \[\sqrt{25-{{x}^{2}}}\]
Here, ‘25’ can be written as $25={{\left( 5 \right)}^{2}}$
Putting the value of ‘25’ in the given expression, we get
$\Rightarrow \sqrt{{{\left( 5 \right)}^{2}}-{{\left( x \right)}^{2}}}$
As we know, ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a+b \right)$
So, by comparing with our expression we get
a=5 and b=x
Performing the required simplification inside the radical, we get
\[\Rightarrow \sqrt{\left( 5+x \right)\left( 5-x \right)}\]
Again, as we know $\sqrt{a\cdot b}=\sqrt{a}\cdot \sqrt{b}$
So, the above expression can be written as
$\Rightarrow \left( \sqrt{5+x} \right)\left( \sqrt{5-x} \right)$
This is the required solution of the given expression.

Note:
‘25’ should be written as the square of ‘5’ to convert the equation to a form where the formula ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a+b \right)$ can be applicable. \[\sqrt{\left( 5+x \right)\left( 5-x \right)}\] can also be the solution of the given expression, but for further simplification the formula $\sqrt{a\cdot b}=\sqrt{a}\cdot \sqrt{b}$ can be used to give the solution as $\left( \sqrt{5+x} \right)\left( \sqrt{5-x} \right)$.