Question

How do you simplify $\dfrac{8+\sqrt{45}}{4}$ ?

Answer 1

Hint: Firstly, solve the root. Dissociate the content inside the root and then use the Law of Radicals to simplify the given expression. The multiplication(dissociative) Lawsuits the most, which is, $\sqrt[n]{{ab}} = \sqrt[n]{a} \times \sqrt[n]{b}$. Now solve the expression by cancelling the common factors in the numerator and the denominator and then express it in simplified form.

Complete step by step solution:
The given expression is, $\dfrac{8+\sqrt{45}}{4}$
Now, let us try simplifying the constants under the root Firstly, and then apply the Multiplication Law to dissociate the terms.
Let us now represent 45 in factor form.
$\Rightarrow \dfrac{8+\sqrt{3\times 3\times 5}}{4}$
Now write the factors in exponential form.
$\Rightarrow \dfrac{8+\sqrt{{{3}^{2}}\times 5}}{4}$
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,
$\Rightarrow \dfrac{8+\sqrt{{{3}^{2}}}\times \sqrt{5}}{4}$
Now cancel the root and the square to simplify further.
$\Rightarrow \dfrac{8+3\sqrt{5}}{4}$
Now dissociate the fraction.
$\Rightarrow \dfrac{8}{4}+\dfrac{3\sqrt{5}}{4}$
Again, cancel the common factors in the numerator and the denominator.
$\Rightarrow 2+\dfrac{3\sqrt{5}}{4}$
Hence, upon simplifying $\dfrac{8+\sqrt{45}}{4}$ we get $2+\dfrac{3\sqrt{5}}{4}$.

Note: The Law of Radicals is derived from the Laws of exponents. The expression $\sqrt[n]{a}$, n is called index, $\sqrt{{ \text{ }}}$ 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.
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.