Question

How do you simplify $3\sqrt {20} + 2\sqrt {45} $?

Answer 1

Hint: Expand the equation first. Use the Law of Radicals to simplify the given expression. The multiplication(dissociative) law suits the most, which is, $\sqrt[n]{{ab}} = \sqrt[n]{a} \times \sqrt[n]{b}$. Try simplifying the constants under the root Firstly, and then apply the Multiplication Law to dissociate the terms. Evaluate further to get a single term.

Formula used:
Laws of radicals, $\sqrt[n]{{ab}} = \sqrt[n]{a} \times \sqrt[n]{b}$

Complete step-by-step answer:
The given expression is, $3\sqrt {20} + 2\sqrt {45} $
According to one of the Laws of radicals which is the multiplication Law of Radicals,
States that $\sqrt[n]{{ab}} = \sqrt[n]{a} \times \sqrt[n]{b}$only if $a,b > 0$.
By using the above reference our expression can be written as,
$ \Rightarrow (3 \times \sqrt {4 \times 5} ) + (2 \times \sqrt {9 \times 5} )$
$ \Rightarrow (3 \times \sqrt 4 \times \sqrt 5 ) + (2 \times \sqrt 9 \times \sqrt 5 )$
reverse the multiplication of the operands.
We can write the respective constant in their square-form as below.
$ \Rightarrow (3 \times \sqrt {{2^2}} \times \sqrt 5 ) + (2 \times \sqrt {{3^2}} \times \sqrt 5 )$
On further simplification, the square and square root gets cancelled.
$ \Rightarrow (3 \times 2 \times \sqrt 5 ) + (2 \times 3 \times \sqrt 5 )$
$ \Rightarrow 6\sqrt 5 + 6\sqrt 5 $
Adding the constants in Infront of the radicals Since the radicands are the same.
$ \Rightarrow 12\sqrt 5 $

$\therefore 3\sqrt {20} + 2\sqrt {45} $ on simplification gives $12\sqrt 5 $.

Additional Information:
The Law of Radicals is derived from the Laws of exponents. The expression $\sqrt[n]{a}$, n is called index,$\sqrt {} $ is called radical, and $a$ is called the radicand. $\sqrt[n]{a} = {a^{\dfrac{1}{n}}}$, the left side of the equation is known as radical form and the right side is exponential form. A Radical represents a fractional exponent in which the numerator and the denominator of the radical contain the power of the base and the index of the radical respectively.

Note:
Memorizing all the Laws of radicals will help to solve any type of same model questions easily. It is important to reduce a radical to its simplest form using the Laws of Radicals for multiplication, division, raising an exponent to an exponent, and taking a radical of a radical makes the simplification process for radicals much easier.