Question

Simplify the following and choose the correct option.
$7\sqrt 5 - 4\sqrt 5 + \sqrt {125} $
A. $3\sqrt 5 $
B. $4\sqrt 5 $
C. $5\sqrt 5 $
D. $8\sqrt 5 $

Answer 1

Hint: We are given an expression with surds, and we need to simplify it by using the concept of addition and subtraction in surds. In surds, we add or subtract the terms outside the root only when the number inside the roots are the same. While adding or subtracting the number inside the root doesn’t change. Using this property, we can simplify the given expression.

Complete answer:
We are given an expression with surds, and we need to simplify it.
In order to simplify this, we just need to use the concept of surds.
Whenever we add or subtract surds the number inside the root always remains the same.
We only add or subtract the numbers outside the root only when the number inside the root is the same.
In our given expression we can see that the number inside the root for the first two terms is the same, that is $5$
So, we can subtract the number outside the root of the second term from the number outside the root of the first term.
$7\sqrt 5 - 4\sqrt 5 + \sqrt {125} = 3\sqrt 5 + \sqrt {125} $
Here in our expression, we can see that the number inside the root is $125$.
So now we need to simplify that in order to proceed with the simplification
We know that we can write $125$ as $25 \times 5$
Substituting this we get,
$7\sqrt 5 - 4\sqrt 5 + \sqrt {125} = 3\sqrt 5 + \sqrt {25*5} $
And we know that the square root of $25$ is $5$ , that is $\sqrt {25} = 5$
Using this we get,
$7\sqrt 5 - 4\sqrt 5 + \sqrt {125} = 3\sqrt 5 + 5\sqrt 5 $
Now we can see that the number inside the root is same in both the terms
So, we can proceed to add the terms outside the root of both the terms
Hence, we get,
$7\sqrt 5 - 4\sqrt 5 + \sqrt {125} = 8\sqrt 5 $

Therefore, the correct option is D

Note: The number outside the roots should be added or subtracted only if the numbers inside the roots are equal. If not, the two terms are just written with the plus or minus sign between them. But this condition does not matter for multiplication or division of surds. During multiplication and division of surds even the number inside the root may change.