Rational Exponents
On Teaching Math
Podcast

Episode 5

Show Notes

 

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Rational Exponents can easily be understood and remembered by students if we expose them to the topic through literacy.  By having students read and understand the notation, they can have instant and clear access to the meaning.

Here’s how to set that up for students.

$3\times 5=3+3+3+3+3$ 

This is calculated, in sequence, as follows.

 

$3\to 6\to 9\to 12\to 15$ 

 

This is fundamentally different than $5\times 3.$ 

 

$5\times 3=5+5+5$ 

This is calculated, in sequence, as follows.

 

$5\to 10\to 15$ 

 

This is an exploration into meaning, not calculation.  Let’s see how the notation and meaning progress to exponents.

 

$\begin{array}{l}3\times 5=3+3+3+3+3\\ 3^5=3\times 3\times 3\times 3\times 3\end{array}$ 

 

In sequence, the calculation is done as follows.

 

$\begin{array}{l}3\times 5=3\to 6\to 9\to 12\to 15\\ 3^5=3\to 9\to 27\to 81\to 243\end{array}$ 

 

Now the pay off.  We need to help students read this: $15\times \frac{1}{5}$.  Division is the reciprocal of multiplication.  Multiplication is repeated addition.  So, division could be thought of as a question. 

$15\times \frac{1}{5}$ can be thought of as a question in math:  What added to itself five times is 15?

 

And $15\times \frac{3}{5}$ is asking, what do you have if you add three of number added to itself five times to get fifteen?  Or, what’s the third step in repeated addition to arrive at 15?

 

Division is a question about repeated addition, the reciprocal of multiplication.  What about a fractional exponent, like ${243}^{1/5}$?

 

Since exponents are repeated multiplication, this is a question about repeated division.  This asks, “What times itself five times is 243?”  Or, “What is the first step in repeated multiplication to arrive at 243 in five steps?”

 

So, ${243}^{3/5}$ asks, “What’s the third step in arriving at 243 with five iterations of repeated multiplication?”  Or, “What is number to the fifth power is 243?  What is that number cubed?”

 

That is why $a^{m/n}=\sqrt[n]{a^m}$ is true.  Let’s look closely.

The number ${64}^{1/2}$ asks what number squared is 64?  That is exactly what $\sqrt{64}$ asks, too.

The last take-away is really about proficiency.  Numbers that are written with rational exponents where students are expected to simplify them, are written to be simplified.  That means, the base is most likely a perfect power of the denominator.

For example, ${64}^{2/3}$ is easily handled because 64 is a perfect cube.  4³ = 64.  As such, the expression can be rewritten and easily simplified.

${64}^{2/3}={\left(4^3\right)}^{2/3}$ 

By multiplying those exponents students end up with 4² or 16.  With a little practice and familiarity with powers of numbers, this is an easy task for students.

Best of all, in my experience anyway, when students learn to read and write radical expressions with indices greater than two, and rational exponents, they remember.  They retain the idea and the procedural proficiencies.