I know this may be a simple question but how does 9^3/2 = (9^1/2)^3? I'm not very math literate.

- anonymous

I know this may be a simple question but how does 9^3/2 = (9^1/2)^3? I'm not very math literate.

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- Mertsj

When you raise something to a power, you multiply the exponents. For example x^4 could be written( x^2)^2 because 2 times 2 is 4.

- Mertsj

So in your example, since 1/2 times 3 = 3/2 those expressions are the same.

- anonymous

Hey Mertsj, thank you for your reply. =) So basically you can just split up the components that make up this expression?

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- Mertsj

yes

- Mertsj

Into factors of the number.

- anonymous

and 'factors' are just like the " (x^2)^2 " of x^4?

- Mertsj

factors are numbers that you multiply. The factors of 12 are 4 and 3 because 4 times 3 = 12

- Mertsj

So
\[x ^{12}=(x ^{4})^{3}=(x ^{6})^{2}\]

- anonymous

So in my example, 3/2. When you raise 1/2 to the third power is it like 1/2 + 1/2 + 1/2 or, 1/2 * 1/2 * 1/2? If I multiply 3 times 1/2 I get 1/2.

- Mertsj

It is like 1/2+1/2+1/2 NOT 1/2*1/2*1/2 because that would be 1/8

- Mertsj

3(1/2)=3/2

- Mertsj

\[3(\frac{1}{2})=\frac{3}{2}\]

- anonymous

Yup, its 1/8. So x^12 = (x^4)^3 = (x^6)^2 that like, 4*3 which equals 12 although when dealing with fractional exponents, its addition?

- Mertsj

No. It's always multiplication.

- anonymous

3(1/2) is 1/2*1/2*1/2 which equals1/8. My bad, I think its actually 3/1*1/2. I am so terrible at math. =(

- anonymous

I think its actually 3/1 * 1/2. You can convert any whole number to a fraction by adding a '1' underneath it I believe.

- Mertsj

it is 3*1/2

- Mertsj

\[(\frac{3}{1})(\frac{1}{2})=\frac{(3)(1)}{(1)(2)}=\frac{3}{2}\]

- anonymous

Thank you for your help. =) It has clarified my confusion. I just have one last question, why would you go through the hassle of converting 9^3/2 to (9^1/2)^3? Is it because you couldn't get the actual root of 9 with 3/2?

- Mertsj

For the simple reason of giving you practice with the idea of multiplying exponents when raising to a power. You probably would not actually do that very often unless you were trying to factor an expression that has fractions for exponents.

- anonymous

So how would you actually break down 9^3/2 to 3^3 in steps without using (9^1/2)^3?

- Mertsj

For example:
\[x ^{\frac{1}{2}}+x ^{\frac{3}{2}}=x ^{\frac{1}{2}}(1+x ^{1})\]

- Mertsj

When you have a fractional exponent, remember that the bottom number of the fraction tells you the root and the top number tells you the power.

- anonymous

I think I get it, 8^2/3 would be 2^2?
This confused me by the way.
x^1/2 + x^3/2 = x^1/2(1 + x^1) Wouldn't that be x^4/2?

- Mertsj

So:
\[9^{\frac{3}{2}}=(\sqrt{9})^{3}=(3)^{3}=27\]

- Mertsj

No. When you multiply, you add the exponents and only like terms can be added.

- Mertsj

And yes, 8^2/3 = 2^2 or 4

- anonymous

I'm still trying to understand this, x^1/2 + x^3/2 = x^1/2(1 + x^1) Its making my brain hurt. =P What do you mean, you add the exponents and only like terms can be added. Yi yi, I'm very sorry.

- anonymous

Wouldn't you distribute x^1/2 to the numbers inside the parentheses of x^1/2 + x^3/2 = x^1/2(1 + x^1)?

- Mertsj

Yes. And in doing so, you would add the exponents. So you would have x^1/2(1) which is x^1/2 and then x^1/2(x^1)=x^1/2(x^2/2)=x^3/2

- anonymous

I keep thinking on x^1/2 + x^3/2, you would add the exponents but that is fractions I believe. A whole other story!

- Mertsj

\[x ^{\frac{1}{2}}(1+x ^{1})=x ^{\frac{1}{2}}(1+x ^{\frac{2}{2}})=x ^{\frac{1}{2}}(1)+x ^{\frac{1}{2}}(x ^{\frac{2}{2}})\]

- Mertsj

And
\[x ^{\frac{1}{2}}(x ^{\frac{2}{2}})=x ^{\frac{1}{2}+\frac{2}{2}}=x ^{\frac{3}{2}}\]

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