Fractional Expontent of a Binomial - Exponent Rules Question

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Aug 7th 2012, 05:44 AM
joseclar

Fractional Expontent of a Binomial - Exponent Rules Question

Hello,

To evaluate the expression (x-2)^1/2 the next steps listed are x^(-2)(1/2)
then x^-1 and finally 1/x. I am fine with the final two steps.

Please help me see how or what rules are used to go from the first to second step. Is this just a basic rules I am forgetting?

So in summary how does (x-2)^1/2 become x^(-2)(1/2)

Thank you.
Aug 7th 2012, 10:28 AM
earboth

Re: Fractional Expontent of a Binomial - Exponent Rules Question

Quote:

Originally Posted by joseclar

Hello,

To evaluate the expression (x-2)^1/2 the next steps listed are x^(-2)(1/2)
then x^-1 and finally 1/x. I am fine with the final two steps.

Please help me see how or what rules are used to go from the first to second step. Is this just a basic rules I am forgetting?

So in summary how does (x-2)^1/2 become x^(-2)(1/2)

Thank you.

1. An example:

$\left(b^3\right)^4 = \left(b^3\right) \cdot \left(b^3\right) \cdot \left(b^3\right) \cdot \left(b^3\right) = \left(b\right)^{3+3+3+3} = b^{3 \cdot 4}$

So if you take a power to a power you have to multiply the exponents.

2. In general:

$\left(b^a\right)^c = b^{a\cdot c} = \left(b^c\right)^a$

3. Apply this rule to your question.
Aug 7th 2012, 12:04 PM
joseclar

Re: Fractional Expontent of a Binomial - Exponent Rules Question

tHANK YOU for the reply. I am aware of the rule for multiplying exponents. My question is how does (x-2)^1/2 become x^-2(1/2). Thanks.
Aug 7th 2012, 12:07 PM
Plato

Re: Fractional Expontent of a Binomial - Exponent Rules Question

Quote:

Originally Posted by joseclar

My question is how does (x-2)^1/2 become x^-2(1/2). Thanks.

Why do you think it does? In what context does it appear?
Aug 7th 2012, 10:00 PM
earboth

Re: Fractional Expontent of a Binomial - Exponent Rules Question

Quote:

Originally Posted by joseclar

tHANK YOU for the reply. I am aware of the rule for multiplying exponents. My question is how does (x-2)^1/2 become x^-2(1/2). Thanks.

... by typo!

Quote:

Originally Posted by joseclar

Hello,

To evaluate the expression (x-2)^1/2 the next steps listed are x^(-2)(1/2)
then x^-1 and finally 1/x. I am fine with the final two steps. <--- who has listed these steps?

Please help me see how or what rules are used to go from the first to second step. Is this just a basic rules I am forgetting?

So in summary how does (x-2)^1/2 become x^(-2)(1/2)

Thank you.

1. I assumed that there was a typo in the original term and that actually was meant $\left(x^{-2} \right)^{\frac12}$

2. If you re-arrange the given term

$(x-2)^{\frac12} = \sqrt{x-2}$

you can see that

$\sqrt{x-2} = \frac1x~\implies~x^3-2x^2-1=0$

has only one real solution at $x = \frac{\sqrt[3]{172-12 \cdot \sqrt{177}}}{6}+\frac{\sqrt[3]{172+12 \cdot \sqrt{177}}}{6} +\frac23$

To answer your question So in summary how does (x-2)^1/2 become x^(-2)(1/2): Only once - and in all other cases this transformation is wrong.