Simplifying a fraction by only multiplying one side by the denominator?

Question

Given the problem: $x\sqrt{3} = 6$

We can solve it this way:

$ x = \frac{ 6 } {\sqrt{3} } \times \frac{ \sqrt{3} } { \sqrt{3} } $

$ x = \frac{ 6 \sqrt{3} } {3} $

$ x = 2 \sqrt{3} $

In the first step, why can we multiply only the right side by $\sqrt{3}$? Shouldn't both sides be modified in the same way to keep them balanced?

Answer 1

You didn’t multiply by $\sqrt{3}$; you multiplied by $$\frac{\sqrt{3}}{\sqrt{3}}=1$$

This is an extremely common trick in simplifying fractions; multiplying by another fraction where the numerator and denominator are the same, because that fraction is $1$. Multiplying sides by $1$ doesn’t change the equality.

Answer 2

To get from here:

$x\sqrt{3} = 6$

to here:

$ x = \frac{ 6 } {\sqrt{3} } \times \frac{ \sqrt{3} } { \sqrt{3} }, $

both sides of the equation were divided by $\sqrt{3},$ not multiplied by $\sqrt{3}.$