Example 3 from Algebra and Trigonometry, 2.6 Other Types of Equations

Solving an Equation Involving Rational Exponents and Factoring

Solve $3 x^{\frac{3}{4}} = x^{\frac{1}{2}} .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
This equation involves rational exponents as well as factoring rational exponents. Let us take this one step at a time. First, put the variable terms on one side of the equal sign and set the equation equal to zero.

$\begin{matrix} 3 x^{\frac{3}{4}} - (x^{\frac{1}{2}}) & = & x^{\frac{1}{2}} - (x^{\frac{1}{2}}) \\ 3 x^{\frac{3}{4}} - x^{\frac{1}{2}} & = & 0 \end{matrix}$

Now, it looks like we should factor the left side, but what do we factor out? We can always factor the term with the lowest exponent. Rewrite $x^{\frac{1}{2}}$ as $x^{\frac{2}{4}} .$ Then, factor out $x^{\frac{2}{4}}$ from both terms on the left.

$\begin{matrix} 3 x^{\frac{3}{4}} - x^{\frac{2}{4}} & = & 0 \\ x^{\frac{2}{4}} (3 x^{\frac{1}{4}} - 1) & = & 0 \end{matrix}$

Where did $x^{\frac{1}{4}}$ come from? Remember, when we multiply two numbers with the same base, we add the exponents. Therefore, if we multiply $x^{\frac{2}{4}}$ back in using the distributive property, we get the expression we had before the factoring, which is what should happen. We need an exponent such that when added to $\frac{2}{4}$ equals $\frac{3}{4} .$ Thus, the exponent on x in the parentheses is $\frac{1}{4} .$

Let us continue. Now we have two factors and can use the zero factor theorem.

$\begin{matrix} x^{\frac{2}{4}} (3 x^{\frac{1}{4}} - 1) & = & 0 \\ x^{\frac{2}{4}} & = & 0 \\ x & = & 0 \\ 3 x^{\frac{1}{4}} - 1 & = & 0 \\ 3 x^{\frac{1}{4}} & = & 1 \\ x^{\frac{1}{4}} & = & \frac{1}{3} & Divide both sides by 3 . \\ {(x^{\frac{1}{4}})}^{4} & = & {(\frac{1}{3})}^{4} & Raise both sides to the reciprocal of \frac{1}{4} . \\ x & = & \frac{1}{81} \end{matrix}$
The two solutions are $0$ and $\frac{1}{81} .$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Solve: ${(x + 5)}^{\frac{3}{2}} = 8.$

How did it go?

Solution

Your turn

Solution

Your turn