Question

How do you simplify \[\sqrt {75} - \sqrt 3 \]?

Answer 1

Hint: In the given expression we need to find the square root value of \[\sqrt {75} \]and \[\sqrt 3 \], then subtract these terms, to solve this expression and we need to combine all the like terms then simplify the terms to get the value as the algebraic expression should be any one of the forms such as addition, subtraction, multiplication and division.

Complete step by step solution:
Given,
\[\sqrt {75} - \sqrt 3 \]
Write 75 as the product of its prime factors i.e.,
\[75 = 5 \times 5 \times 3 = {5^2} \times 3\]
Now, we have
\[\sqrt {75} - \sqrt 3 \]
As, we have 75 as the product of its prime factors, hence substitute it as:
\[ \Rightarrow \sqrt {{5^2} \times 3} - \sqrt 3 \]
Find, any possible roots as:
\[ \Rightarrow 5\sqrt 3 - 1\sqrt 3 \]
Now, we have like terms, hence we get:
\[ \Rightarrow \left( {5 - 1} \right)\sqrt 3 \]
\[ \Rightarrow 4\sqrt 3 \]
\[ \Rightarrow 6.92\]
Therefore,
\[\sqrt {75} - \sqrt 3 = 6.92\].

Additional information:
There are two keys to combining radicals by addition or subtraction: look at the index, and look at the radicand. If these are the same, then addition and subtraction are possible. If not, then you cannot combine the two radicals.
To multiply radicals, the roots of the radical expressions are the same, you can use the Product Raised to a Power Rule to multiply and simplify the expressions both inside and outside the radical by multiplying. Multiply all numbers and variables inside the radical together.

Note: To add or subtract radicals, the indices and what is inside the radical (called the radicand) must be exactly the same. If the indices and radicands are the same, then add or subtract the terms in front of each like radical and if the indices or radicands are not the same, then you cannot add or subtract the radicals.