Powers

[davehogan]

My calculator tells me that 4^(3/2)=8, but I don't understand how this is done mathematically. Can anyone explain it for me, please?

 

[pka]

My calculator tells me that 4^(3/2)=8, but I don't understand how this is done mathematically. Can anyone explain it for me, please?

\(\displaystyle \left( {{4^{\frac{3}{2}}}} \right) = {\left( {\sqrt 4 } \right)^3}\)

 

[JeffM]

My calculator tells me that 4^(3/2)=8, but I don't understand how this is done mathematically. Can anyone explain it for me, please?

Let's consider the meaning of a power that is a positive integer.

4^3 = 4 * 4 * 4 = 64. Four is the base in this case, and three is the power. We simply multiply the base times itself three times.

4^3 * 4^4 = (4 * 4 * 4) * (4 * 4 * 4 * 4) = 4^7.

If you multiply two numbers that are both powers of the same base, then the answer is a number with the same base and a power that is the sum of the two powers.

By that logic, 64 = 4^3 = 4^1 * 4^2. But 4^2 = 4 * 4 = 16. And 4 * 16 = 64. So 4^1 = 4.

Any number to the power of one is itself.

So two of the laws of powers are:

\(\displaystyle a^b * a^c = a^{(b + c)}\ and\ a^1 = a.\) With me so far?

Now what happens if we take a power to a power.

(2^2)^3 = 2^2 * 2^2 * 2^2 = 2^6. Oh my, the powers multiply.

Another law of powers is:

\(\displaystyle \left(a^b\right)^c = a^{(c * b)}.\)

Now consider the multiplicative inverses of the positive integers. The multiplicative inverse of 2 is is 1/2.

\(\displaystyle If\ u \ne 0,\ the\ multiplicative\ inverse\ of\ a = \dfrac{1}{a}. \)

What does it mean if we say a = 49^(1/2)? Well, we know powers add when we multiply powers with the same base.

So \(\displaystyle a \ge 0\ and\ a * a = 49^{(1/2)} * 49^{(1/2)} = 49^{\{(1/2) + (1/2)\}} = 49^1 = 49.\)

But that means in this case that \(\displaystyle a = \sqrt{49} = 7.\)

So another law of powers is: \(\displaystyle a \ge 0\ and\ b > 0 \implies a^{(1/b)} = \sqrt{a}.\)

Putting all that together, we get what pka already told you.

\(\displaystyle 4^{(3/2)} = 4^{\{3 * (1/2)\}} = \left(4^{(1/2)}\right)^3 = \left(\sqrt{4}\right)^3 = 2^3 = 2 * 2 * 2 = 8.\)

 

[davehogan]

I have never before had such an informative reply. Thank you very much.

 

[HallsofIvy]

It is useful to know that \(\displaystyle a^{m/n}= (a^{1/n})^m= (a^m)^{1/n}\). So \(\displaystyle 4^{3/2}\) can be done as \(\displaystyle (4^{1/2})^3= 2^3= 8\) or as \(\displaystyle (4^3)^{1/2}= 64^{1/2}= 8\).

 

[HallsofIvy]

It is useful to know that \(\displaystyle a^{m/n}= (a^{1/n})^m= (a^m)^{1/n}\). So \(\displaystyle 4^{3/2}\) can be done as \(\displaystyle (4^{1/2})^3= 2^3= 8\) or as \(\displaystyle (4^3}^{1/2}= 64^{1/2}= 8\(\displaystyle .\)\)