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What is \[\sqrt {126} + \sqrt {56} \]in the standard form?

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Last updated date: 13th Jul 2024
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Answer
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Hint: To write the given expression in standard form, we need to simplify both the terms in the expression first. For that, first we need to prime factorise the numbers and then write the numbers which can be written as squares in the form of squares of that number and then find their square root and write their square root in simplified form. We know,
\[x \times x = {x^2}\]
\[\sqrt {{x^2}} = |x|\]
So, in this question, when we square root the prime factors, we will use the above formula and then multiply it with the remaining terms. Also, we know
\[\sqrt {xy} = \sqrt x \sqrt y \]
\[x\sqrt y + z\sqrt y = (x + z)\sqrt y \]
\[x\sqrt y + w\sqrt z \] cannot be added.
And, we will use this to add the simplified form of the two numbers.

Complete step-by-step solution:
We need to find \[\sqrt {126} + \sqrt {56} \]in standard form.
For that, we will simplify \[\sqrt {126} \]and \[\sqrt {56} \]separately and then add them.
Prime Factorisation \[126\], we get,
\[126 = 2 \times 3 \times 3 \times 7\]
\[126 = 2 \times {3^2} \times 7\]
Now, square rooting both the sides,
\[\sqrt {126} = \sqrt {2 \times {3^2} \times 7} \]
\[\sqrt {126} = \sqrt {2 \times 7} \times \sqrt {{3^2}} \] (Separating the square terms)
\[\sqrt {126} = \sqrt {2 \times 7} \times |3|\]
\[\sqrt {126} = \sqrt {14} \times 3\]
\[\sqrt {126} = 3\sqrt {14} \] ----(1)
Now, simplifying \[\sqrt {56} \]
Prime Factorisation \[56\], we get,
\[56 = 2 \times 2 \times 2 \times 7\]
\[56 = {2^2} \times 2 \times 7\] (Writing \[2 \times 2\] as a \[{2^2}\])
Square rooting both sides,
\[\sqrt {56} = \sqrt {{2^2} \times 2 \times 7} \]
\[\sqrt {56} = \sqrt {{2^2}} \times \sqrt {2 \times 7} \] (Separating the square term)
\[\sqrt {56} = |2| \times \sqrt {2 \times 7} \]
\[\sqrt {56} = 2 \times \sqrt {14} \]
\[\sqrt {56} = 2\sqrt {14} \] -----(2)
Now, from (1) and (2), we get,
\[\sqrt {126} + \sqrt {56} = 3\sqrt {14} + 2\sqrt {14} \]
\[\sqrt {126} + \sqrt {56} = 5\sqrt {14} \]
Hence, \[\sqrt {126} + \sqrt {56} = 5\sqrt {14} \]in standard form.

Note: While solving, we need to keep in mind that the terms which cannot be written as squares will remain in under root and the ones which can be written as squares will be simplified. And, we need to make sure that terms in square root can be added and what terms cannot be. In these types of questions, when we take square root, we usually don’t take negative signs but only positive signs.