Can you Explain the Mysteries of Exponents?

[musicgold]

Hi,

I am helping my kid with exponents. I told her that the exponent tells us how many times we should multiply the base number. While it works with a simple example like 4^6, I am not sure how to explain her why 4^0 =1 and why 27^(1/3) = 3.

Any ideas?

Thanks.

 

[ehild]

Show how to multiply and divide powers of the same base -with simple numbers first.

For example, $2^3\cdot 2^4= (2\cdot2\cdot2)\cdot(2\cdot2\cdot2\cdot2) = 2^7$. The exponents add. aⁿ*a^m=a^n+m.

You can simplify the fraction when dividing powers.

$$\frac{2^4}{2^3}=\frac{2\cdot2\cdot2\cdot2}{2\cdot2\cdot2}=2= 2^{4-3}$$.
$$\frac{a^n}{a^m}=a^{n-m}$$



What happens if n=m?

$$\frac{2^3}{2^3}=\frac{2\cdot2\cdot2}{2\cdot2\cdot2}=1= 2^{3-3}=2^0$$

a⁰=1...

You can show what a negative exponent means:

$$\frac{2^3}{2^4}=\frac{2\cdot2\cdot2}{2\cdot2\cdot2\cdot2}=\frac{1}{2}= 2^{3-4}=2^{-1}$$.


The next is to show how to get the power of a power:

$\left(2^3\right)^4=(2\cdot2\cdot2)\cdot(2\cdot2\cdot2)\cdot(2\cdot2 \cdot2)\cdot(2\cdot2\cdot2)= 2^{12}$ You multiply the powers. (aⁿ)^m=a^nm.

What does it mean when the power is a fraction, 1/3, for example?

$\left(2^{1/3}\right) ^3=2^{\frac{1}{3}\cdot 3}=2^1=2$.

a^1/3 is a number the third power of which is a.

As for 27^1/3: 27=3³.
27^1/3=(3³)^1/3=3^3*1/3=3¹=3

ehild

 

[micromass]

It all relies on knowing the laws $a^{m+n} = a^m a^n$ This is a fundamental property for understanding exponents.

Then it is certainly true that $1+0 = 1$. So if we put those in the exponents, then it must be true that $4^{1 + 0} = 4^1$. Thus $4^1 4^0 = 4^1$. Of course, $4^1 = 4$. Thus we have something like $4\cdot 4^0 = 4$. So $4^0$ is some number when multiplied by $4$, it will give $4$ again. We see immediately that $4^0 = 1$.

For $27^{1/3}$ something similar holds. Of course we know that $\frac{1}{3}+ \frac{1}{3} + \frac{1}{3} = 1$. So if we put this in the exponents, we get
$$27^{\frac{1}{3}+ \frac{1}{3} + \frac{1}{3}} = 27^1 = 27$$
And when using our fundamental property, we see that
$$27^{1/3}27^{1/3}27^{1/3} = 27$$
or just
$$(27^{1/3})^3 = 27$$
So $27^{1/3}$ is the number such that if we cube it, we get $27$. But by inspection we see that $3$ is such a number since $3^3 = 27$, so we must have $27^{1/3} = 3$.

 

[musicgold]

Thanks folks.