Question

What is the logical first step in rationalizing the denominator of $\frac{3}{1+\sqrt{x}}$?

Short answer

The first step in rationalizing the denominator of the given expression is to multiply the denominator of the given expression by $1$.

Step-by-step solution

Step 1. State steps to rationalize a denominator.

The steps to rationalize a denominator are:

Multiply the given quantity by 1, so that it cannot change the value of given quantity.

Now if the denominator is in the form of$1+\sqrt{a}$ then multiply the numerator as well as the denominator with$1-\sqrt{a}$ and convert the given quantity into no radical form,

Step 2. Write down the given expression.

The given term is,

$\frac{3}{1+\sqrt{x}}$

Multiply expression by $1$.

$\begin{array}{c}\frac{3}{1+\sqrt{x}}=\frac{3}{1+\sqrt{x}}\times 1\\ =\frac{3}{1+\sqrt{x}}\end{array}$

Step 3. Rationalize the expression.

The denominator is of the form $1+\sqrt{a}$. So,

$\begin{array}{c}\frac{3}{1+\sqrt{x}}=\frac{3}{1+\sqrt{x}}\times \frac{1-\sqrt{x}}{1-\sqrt{x}}\\ =\frac{3\left(1-\sqrt{x}\right)}{\left(1^2-{\left(\sqrt{x}\right)}^2\right)}\\ =\frac{3\left(1-\sqrt{x}\right)}{1-x}\end{array}$

Thus, the first step in rationalizing the denominator of the given expression is to multiply and divide the given expression by $1$.