Rational Exponents

Rational exponents go by many names, including the more specific phrase "fractional exponents" and our personal favorite: "Radicals." But no matter what we call them, rational exponents are very important in the world of math. And even though these exponents are rational, they can behave somewhat irrationally. Whether you're feeling confused about rational exponents or you're encountering this concept for the first time, you're in the right place. But what exactly are rational exponents? How do they work? Let's find out:

What is a rational exponent?

The phrase "fractional exponent" tells us what these numbers are. These are numbers that include exponents in the form of fractions. Examples include:

$4^{\frac{1}{2}}, 1^{\frac{1}{3}}, 7^{\frac{5}{8}}$ , and so on.

You're probably already familiar with whole-number exponents. For example, we know that when an exponent is a whole number it means repeated multiplication of the base:

$8^{2} = 8 \times 8 = 64$

$8^{3} = 8 \times 8 \times 8 = 64$

But that method becomes more difficult with rational exponents, and we must use a different system to solve these numbers:

How to figure out rational exponents

So what exactly is the "rule" for solving rational exponents?

Let's start with an easy example:

$4^{\frac{1}{2}}$

This behaves in a manner opposite to whole-number exponents. Instead of multiplying four with itself, we take the square root of four.

In other words:

$4^{\frac{1}{2}} = \sqrt{4}$

$\sqrt{4} = 2$

Which is asking the question "what number, times itself is equal to 4?"

Seems easy, right? But what happens if we use a slightly more complicated rational exponent?

$4^{\frac{1}{3}} = \sqrt[3]{4}$ . We call this the "cubic root" of four.

Which is asking the question, "what number times itself 3 times is equal to 4?"

$4^{\frac{1}{4}} = \sqrt[4]{4}$ . This is the fourth root.

This pattern continues indefinitely into the fifth, sixth, seventh roots and beyond.

Why does the rule for rational exponents work?

If you're feeling a little suspicious about this new rule, don't worry -- we're about to demonstrate that it makes sense.

It all comes back to the laws of exponents:

This law tells us that when we multiply exponents with the same coefficients, we simply add them together.

For example:

$y^{4} y^{3} = (y^{4}) (y^{3}) = (y y y y) (y y y) = (y y y y y y y) = y^{7}$

In other words, $y^{4} y^{3} = y^{(4 + 3)} = y^{7}$

But what does this have to do with square roots and rational exponents?

Let's apply the laws of exponents to rational numbers:

$9^{\frac{1}{2}} 9^{\frac{1}{2}} = 9^{(\frac{1}{2} + \frac{1}{2})} = 9^{1} = 9$

That's right -- two halves make a whole, and $9^{\frac{1}{2}} 9^{\frac{1}{2}} = 9$

When we multiply a number by itself to get another number, we know that the first number is the square root.

In other words, $9^{\frac{1}{2}} = \sqrt{9}$

The same rule applies even with more complex rational exponents:

$7^{\frac{1}{8}} = 7^{\frac{1}{8}} \times 7^{\frac{1}{8}} \times 7^{\frac{1}{8}} \times 7^{\frac{1}{8}} \times 7^{\frac{1}{8}} \times 7^{\frac{1}{8}} \times 7^{\frac{1}{8}}$

$= 7^{1}$

$= 7$

In other words, $7^{\frac{1}{8}}$ is the eighth root of 7.

The formula for rational exponents

Like most math rules, we can put what we've just learned into a neat formula:

$x^{\frac{1}{n}}$ is the n-th root of x

Plug in a few values, and we might get something like this:

$27^{\frac{1}{3}} = \sqrt[3]{27}$

We can verify that $3 \times 3 \times 3 = 27$ , so the answer is:

$\sqrt[3]{27} = 3$

Getting complicated with rational exponents

You might have noticed that we've only covered simple fractions for exponents so far, such as $5^{\frac{1}{4}}$ . But what happens when our exponents become more complex? For example:

$5^{\frac{4}{5}}$

Let's break down this number. We can rewrite the exponent as a sum of a whole number and a fraction: $\frac{4}{5} = 1 + \frac{1}{4}$ . In general, for a rational exponent m/n, where m and n are integers, we can express it as a whole number and a fraction:

$\frac{m}{n} = a + \frac{b}{n}$ where a and b are whole numbers.

This allows us to rewrite the expression with the exponent:

$x^{\frac{m}{n}} = x^{a + \frac{b}{n}}$

Using the properties of exponents, we can separate the expression into two parts:

$x^{\frac{m}{n}} = x^{a} \times x^{\frac{b}{n}}$

Now, we can apply the rules for fractional exponents:

$x^{\frac{m}{n}} = x^{a} \times {x^{b}}^{\frac{1}{n}}$

When confronted with complicated fractional exponents, you have two options:

1. Raise x to the power of a (the whole number), and then take the nth root of the result.

2. Take the nth root of x, and then raise the result to the power of a (the whole number).

In our example, we can compute $5^{\frac{5}{4}}$ using either of these options:

$5^{\frac{5}{4}} = 5^{1} \times \sqrt[4]{5^{1}} = 5 \times \sqrt[4]{5}$

$5^{\frac{5}{4}} = \sqrt[4]{5^{1}} = \sqrt[4]{5}$

Both methods give the same result: $5^{\frac{5}{4}} = 5 \times \sqrt[4]{5}$ .

Examples of complicated rational exponents

If we wanted to simplify $27^{\frac{2}{3}}$ , we would do the following:

$27^{\frac{2}{3}} = {\sqrt[3]{27}}^{2} = 3^{2} = 9$

If we wanted to simplify $4^{\frac{3}{2}}$ , we would do the following:

$4^{\frac{3}{2}} = \sqrt{4^{3}} = \sqrt{4 \times 4 \times 4} = \sqrt{64} = 8$

What about $27^{\frac{4}{3}}$ ? Easy:

$27^{\frac{4}{3}} = {3 \times \sqrt[3]{27}}^{4} = 3^{4} = 81$

Note that we simplified the two latter rational exponents in slightly different ways. Depending on whether we prefer to solve square roots or whole-number exponents, we can choose the strategy that we like best. Either way, we will come to the same conclusion!

Real-world examples of rational exponents

But when would we ever need to use rational exponents to solve real-world problems? Here are a few examples:

The inflation of real estate can be calculated with the equation:

$i = {(\frac{p_{2}}{p_{1}})}^{\frac{1}{n}} - 1$

where i represents the interest, $p_{1}$ is the starting price, $p_{2}$ is the end price, and n is the time in the number of years.

We can also use the following formula to calculate compound interest:

$F = P (1 + i) n$

where F represents the future value, P is the present value, i is the interest rate, and n is the number of years. Note that the number of years can be written as a fraction, such as 18 out of 12 months or $\frac{3}{2}$ .

As you can see, rational exponents can be very useful in the financial world.

Topics related to the Rational Exponents

nth Roots

Radical Equations

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Flashcards covering the Rational Exponents

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College Algebra Flashcards

Practice tests covering the Rational Exponents

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College Algebra Diagnostic Tests

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