Examples: Perfect Fractional Exponents


In math, we often learn one set of facts through calculation, and another through related calculations. For example 5 \times 5 = 25. We can calculate this one. The related fact is that \sqrt{25}=5. We know this fact through association rather than calculation. Later, when solving algebraic equations involving a square, we will also need to remember that -5 \times -5 = 25.

Or how about this, 2 \times 2 \times 2 \times 2 \times 2 = 32. The associated number fact is that, \sqrt[5]{32}=2. We don’t ‘calculate’ that the number multiplied by itself 5 times over to make 32 is 2: we know it from memory.

To help with remembering exponent facts, here is a short list of most commonly used powers: Exponent Cheat Sheet. If you don’t already recall these facts easily, keep the reference sheet handy so that you can focus on the concepts rather than the arithmetic. Here is a long list from wikipedia.

Evaluating positive fractional exponents

Generally, a fractional exponent leads to an irrational number. These questions are called ‘perfect’ because the answers are all integers, and can be evaluated without using a calculator.

Example1:  16^{\frac{1}{4}}

We’re asking, what power of four is equal to 16? This is the part that we recall. (For the fourth root you can also do this little trick – square root then square root):

    \[16^{\frac{1}{4}}=\big(16^{\frac{1}{2}}\big)^{\frac{1}{2}}=4^{\frac{1}{2}}=2\]

Or just check the Exponent Cheat Sheet.

Example 2: 16^{\frac{3}{4}}

First, we need to know that 16^{\frac{1}{4}}=2. Without knowing that, this question is simply really hard.

This is how we handle it:

We know that \frac{3}{4}= \frac{1}{4} \times 3.

And we know that (a^m)^n=a^{m\times n}

This is why we can rewrite 16^{\frac{3}{4}} as \big(16^{\frac{1}{4}}\big)^3

In this form, we first evaluate the bracket from memory then calculate the power:

    \[\big(16^{\frac{1}{4}}\big)^3=\big(2\big)^3=8\]

Example 3: 125^{\frac{2}{3}}

We mean, take the cube root of 125 then square it.

    \begin{align*}125^{\frac{2}{3}} \\[10pt]&= \big(125^{\frac{1}{3}}\big)^2 \\[10pt]&= \big(5\big)^2\\[10pt]&=25\end{align*}

Example 4: 64^{\frac{3}{2}}

We mean, take the square root of 64 then cube.

    \begin{align*}64^{\frac{3}{2}} \\[10pt]&= \big(64^{\frac{1}{2}}\big)^3 \\[10pt]&= \big(8\big)^3\\[10pt]&=512\end{align*}

Example 5:4096^{\frac{1}{6}}

Now, 4096^{\frac{1}{6}} isn’t on the cheat sheet. So let’s look for a likely relative. We find:

    \[1024=2^{10}\]

Now,

    \[4096 = 1024 \times 2 \times 2 = 2^{10} \times 2 \times 2 = 2^{12}\]

And,

    \[2^{12}=\big(2^2\big)^6=4^6\]

Therefore, 4096=4^6.

Now we have it, 4096^{\frac{1}{6}}=4.

Evaluating negative exponents

Division can be expressed using \div or by using a fraction or by using a negative exponent. A negative exponent does not make the value negative because dividing does not make a value negative.

1 divided by 7 can be written as:

    \[1 \div 7 = \frac{1}{7}=7^{-1}\]

Note that all of these are positive (all are equal to 0.142\dots). Generally the first thing we do with a negative exponent (which means ‘divide’) is to write it as a fraction (another way to say ‘divide’).

Example 6: 2^{-3}

A negative exponent represents repeated division. When we see x in algebra, we understand that it means 1\cdot x, but the 1 is not necessary to write. Here, it can be helpful to write the 1, as seen in the second line:

    \begin{align*}2^{-3}\\[10pt]&=1\times 2^{-3}\\[10pt]&=1 \div 2^3\\[10pt]&=\frac{1}{2^3}\\[10pt]&=\frac{1}{8}\end{align*}

 

Example 7: 25^{-\frac{3}{2}}

    \begin{align*}25^{-\frac{3}{2}}\\[10pt]&=\frac{1}{25^\frac{3}{2}}\\[10pt]&=\frac{1}{\big(25^\frac{1}{2}\big)^3}\\[10pt]&=\frac{1}{5^3}\\[10pt]&=\frac{1}{125}\end{align*}

Fractions with Negative Exponents

In examples 8, 9 and 10 we take a negative power of a fraction. To understand the process, you need to remember how to divide a fraction. Here’s a reminder:

 

    \begin{align*}\frac{3}{4}\div \frac{7}{11}\\[10pt]&=\frac{3}{4}\times\frac{11}{7} \\[10pt]&=\frac{33}{28}\end{align*}

Example 8: \Big(\frac{4}{5}\Big)^{-2}

Watch what happens to the fraction when there is a negative power.

    \begin{align*}\big(\frac{4}{5}\big)^{-2}\\[10pt]&=\frac{1}{\frac{4^2}{5^2}}\\[10pt]&=1\div \frac{4^2}{5^2}\\[10pt]&=\frac{1}{1}\times\frac{5^2}{4^2}\\[10pt]&=\frac{5^2}{4^2}\\[10pt]&=\frac{25}{16}\end{align*}

Notice that our orignial fraction flips when the exponent is negative. Let’s write that down in general:

In general,

    \[{\Big(\frac{a}{b}\Big)^{-n}=\Big(\frac{b}{a}\Big)^n}\]

Example 9: \Big(\frac{1000}{343}\Big)^{-\frac{1}{3}}

    \begin{align*}\Big(\frac{1000}{343}\Big)^{-\frac{1}{3}}\\[10pt]&=\Big(\frac{343}{1000}\Big)^{\frac{1}{3}}\\[10pt]&=\frac{343^{\frac{1}{3}}}{1000^{\frac{1}{3}}}=\frac{7}{10}\end{align*}

Example 10: \big(\frac{125}{8}\big)^{-\frac{2}{3}}

    \begin{align*}\Big(\frac{125}{8}\Big)^{-\frac{2}{3}}\\[10pt]&=\Big(\frac{8}{125}\Big)^{\frac{2}{3}}\\[10pt]&=\frac{8^{\frac{2}{3}}}{125^{\frac{2}{3}}}=\frac{4}{25}\end{align*}

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