Question

How do you simplify (7 square roots of 5) + (the square root of 50)?

Answer 1

Hint: Assume the mathematics expression that will be formed by the given statement as ‘X’. Now, consider the line before the plus sign and multiply 7 with the square root of 5 to form half part of X. Now, consider the line after the plus sign and take the square root of 50 to form the other half part. To simplify the obtained expressions use the prime factorization method to find the square root of 50.

Complete step by step solution:
Here, we have been provided with the statement: (7 square root of 5) + (the square root of 50) and we are asked to simplify it. That means we have to find the simplified mathematical expression for this statement.
Now, we have two parts in the statement separated by a plus sign. Let us assume the mathematical expression that will be formed as ‘X’. So, we have,
\[\Rightarrow \] X = (7 square roots of 5) + (the square root of 50)
Now, let us consider the first part of the expression. So, we have the statement ‘7 square roots of 5’, that means 7 times square root of 5. This can be written mathematically as: -
\[\Rightarrow X=7\times \sqrt{5}+\] (the square root of 50)
\[\Rightarrow X=7\sqrt{5}+\] (the square root of 50)
Now, let us consider the second part of the expression. So, we have the statement ‘the square root of 50’, that can be written mathematically as: -
\[\Rightarrow X=7\sqrt{5}+\sqrt{50}\]
Here, we need to simplify the term \[\sqrt{50}\]. To do this we will use the prime factorization method and try to form a group of identical prime factors such that in exponential form the exponent of these prime factors becomes 2. So, we can write 50 as: -
\[\Rightarrow 50=2\times 5\times 5\]
In exponential form, we can write the above expression as: -
\[\Rightarrow 50=2\times {{5}^{2}}\]
Substituting this value in the expression X, we have,
\[\begin{align}
  & \Rightarrow X=7\sqrt{5}+\sqrt{2\times {{5}^{2}}} \\
 & \Rightarrow X=7\sqrt{5}+5\sqrt{2} \\
\end{align}\]
Hence, the above expression is our answer.

Note: You may note that the expression that we have obtained in the end cannot be simplified further. However, we can find the decimal representation of \[\sqrt{5}\] and \[\sqrt{2}\] using the long division method and get the answer in decimal form but it is not necessary here as we aren’t asked to do so. You must remember the steps of finding the prime factors of a number because it is used to find the square root and cube root of large numbers.