Examples: Perfect Fractional Exponents


Let’s consider the set of numbers known as perfect squares once again. These numbers are the areas of squares of side length 1, 2, 3 .. and so on.

Here are the first 12 positive perfect squares:

    \[ \boxed{1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144} \]

If you are familiar enough with these values, then it is easy for you to conclude facts such as:

\sqrt{81}=9 and \sqrt{49}=7

and so on.

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 helpful exponent facts, here is a short list of most commonly used powers: Exponent Cheat Sheet. It is always helpful in math to be able to recall some number relations, so that the teacher can use these number relations when explaining new and complex concepts – if the concepts are difficult, we like to keep the arithmetic simple. Here is a long list from wikipedia. Just keep looking at one of these sheets for answers until you find that you’ve used the number facts so much you know them.

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 216^{\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 464^{\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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