Question

The value of \[3\sqrt {28} + 2\sqrt 7 \] is equal to
A) \[8\sqrt 7 \]
B) \[4\]
C) \[2\sqrt 7 \]
D) \[7\]

Answer 1

Hint: Here in this question, we have to find the value of given radical expression. First convert \[\sqrt {28} \] to \[\sqrt 7 \] by using the multiplication of 7. Then by using the properties of square roots and radicals and on further simplification we get required value which is in simplest form, that cannot be simplified further.

Complete step-by-step solution:
Operations with Square Roots. You can perform a number of different operations with square roots. Some of these operations involve a single radical sign, while others can involve many radical signs. The some properties or rules can be applied.
Consider the given expression
\[ \Rightarrow \,\,3\sqrt {28} + 2\sqrt 7 \]-----------(1)
28 is the multiple of 7, then
 \[ \Rightarrow \,\,3\sqrt {7 \times 4} + 2\sqrt 7 \] -------(2)
By the property, whenever we have two or more radical terms which are multiplied with same index, then we can put only one radical and multiply the terms inside the radical. i.e., \[\sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{{a \times b}}\]
then equation (2) becomes
\[ \Rightarrow \,\,3 \cdot \sqrt 4 \times \sqrt 7 + 2\sqrt 7 \]
4 is the square number of 2, then
\[ \Rightarrow \,\,3 \cdot \sqrt {{2^2}} \times \sqrt 7 + 2\sqrt 7 \] --------(3)
By another property, when a number is multiplied by itself, the product is called the square of that number. The number itself is called the square root of the product. i.e., \[\sqrt {a \times a} = \sqrt {{a^2}} = a\]
Then, equation (3) becomes
\[ \Rightarrow \,\,3 \cdot 2 \times \sqrt 7 + 2\sqrt 7 \]
\[ \Rightarrow \,\,6\sqrt 7 + 2\sqrt 7 \]
On simplification, we get
\[ \Rightarrow \,\,8\sqrt 7 \]

Hence, the value of \[3\sqrt {28} + 2\sqrt 7 \] is option (A) \[8\sqrt 7 \] is the correct answer.

Note: In mathematics we have many kinds of numbers namely, natural numbers are the counting numbers, whole numbers are the natural numbers along with zero, integers are the whole numbers and the also include the negative of natural numbers, rational numbers are in the form of p by q form, irrational numbers and real numbers.