Question

Simplify the expression, $\sqrt {49 - {x^2}} $.

Answer 1

Hint: Whenever we are dealing with roots, it is best simplified by assigning the expression to a variable, say “y”. But different expressions need different approaches to be simplified. Sometimes one might have to deal with questions that cannot be simplified. In that case you can re-express the expression so as to ease the calculations.

Complete step-by-step solution:
The expression in the question cannot be simplified further. However, it can be re-expressed.
49 is the square of 7. So, we can write $\sqrt {49 - {x^2}} = \sqrt {{7^2} - {x^2}} $.
We know that, ${a^2} - {b^2} = (a + b)(a - b)$.
Using this algebraic identity, we can rewrite the given expression as
$\sqrt {49 - {x^2}} = \sqrt {{7^2} - {x^2}} = \sqrt {(7 + x)(7 - x)} $….. equation (i)
In the above expression we have clearly taken $a = 7,b = x$and implemented the algebraic identity, ${a^2} - {b^2} = (a + b)(a - b)$.
Now we will use the product property of radicals, where \[a \geqslant 0\;or\;b \geqslant 0\],$\sqrt {a \times b} = \sqrt a \times \sqrt b $to again re-express the equation (i).
$\sqrt {49 - {x^2}} = \sqrt {{7^2} - {x^2}} = \sqrt {(7 + x)(7 - x)} = \sqrt {(7 + x)} \times \sqrt {(7 - x)} $

Thus, the final answer is $\sqrt {49 - {x^2}} = \sqrt {{7^2} - {x^2}} = \sqrt {(7 + x)(7 - x)} = \sqrt {(7 + x)} \times \sqrt {(7 - x)} $

Note: There are a few things to be kept in mind when simplifying radicals. If there are no perfect squares under the radical, simplification is not possible. For fractions under radical, it is considered simplified only if the denominator is no more radical. For radicals in the denominator, rationalizing of the same is required. Simplification of an expression doesn’t mean the final result will change. The result remains the same, though the look of the expression changes. Simplification is used to avoid cumbersome calculations and for easy deciphering of how the solution is proceeded with.