It’s hard to teach square roots. Heck – it’s hard to learn square roots!

Multiplication is the shortcut for addition. And exponents are the shortcut for multiplication.

So we should be able to play similar games with exponents as we do with multiplication.

And we can!

Square roots are exponents.

What is a square root? It’s a fractional exponent! Like this:

$latex \sqrt9 = 9^\frac{1}{2}$

If you convert all radicals to fractional exponents then you can manipulate them using the standard rules of exponents.

I taught this for the first time last semester. And the students loved it.

When I would ask, “Do you want me to work it with the radical, or convert it?” I got a unanimous “Convert it!”

Teach square roots with the concept of “one half.”

You have 10 cupcakes. You want to split them evenly with me. YUM!

So the number of cupcakes you get, added to the (equal) number of cupcakes I get, is 10. We each get 5 cupcakes because $latex 5+5=10$. And we write: $latex \frac{1}{2}\times10 = 5$.

Now, let’s do it with magical cupcakes. How about we “split” 9 magical cupcakes evenly – but in a special freakish way (after all, they ARE magical cupcakes).

This time we split it so the amount of cupcakes you get multiplied with the (equal) amount of cupcakes I get is 9.

So we each get 3 magical cupcakes, because $latex 3 \times 3 =9$.  And don’t worry – 3 magical cupcakes are just as good as 4.5 regular cupcakes.

And for this magical splitting, we write $latex 9^\frac{1}{2} = 3$.

The concept of halving is there!

Just as half of 10 is 5, the “special halving” of 9 is 3.

Which means we can do the same with cube roots (raised to the $latex \frac{1}{3}$), fourth roots (raised to the $latex \frac{1}{4}$), etc.

Some quick examples:

$latex 125^\frac{1}{3}=5$ because 125 magically split into “thirds” is 5. (Remember, we’re reconstituting 125 using multiplication: $latex 5 \times 5 \times 5 = 125$).

$latex 81^\frac{1}{4}=3$ because “one quarter” of 81 is 3 – in our special multiplication sense.

So how about it?

Can you extend the concept of fractions and splitting in your classes? Will it help you students?

Share your thoughts in the comments – and don’t forget to tweet it too!

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