Fractional Expontent of a Binomial - Exponent Rules Question

[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

 

[srmichael]

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

Is (x-2)^1/2 supposed to be [x^(-2)]^(½)? That's what it seems like based on the next step. The second step is utilizing this property: [x^m]^n = x^(mn)

Sorry, but LaTex is acting up on me this morning.

 

[joseclar]

\(\displaystyle Let\ x = 2.\)

\(\displaystyle (x - 2)^{1/2} = \sqrt{x - 2} = \sqrt{0} = 0 \ne \dfrac{1}{2}.\)

\(\displaystyle SO\ IT\ IS\ FALSE\ THAT\ (x - 2)^{1/2} = \dfrac{1}{x}.\)

HOWEVER

\(\displaystyle (x^{-2})^{1/2} = x^{[(-2) * (1/2)]}\ by\ one\ of\ the\ laws\ of\ exponents.\)

\(\displaystyle So\ (x^{-2})^{1/2} = x^{[(-2) * (1/2)]} = x^{(-1)} = \dfrac{1}{x}.\)

Thanks for the response.

Again, I am fine with how you multiply exponents and use negative exponents. My question is what law/property allows you to make (x-2)
x^-2? That is the only step I want clarification on.

 

[srmichael]

Thanks for the response.

Again, I am fine with how you multiply exponents and use negative exponents. My question is what law/property allows you to make (x-2)
x^-2? That is the only step I want clarification on.

x-2 in no way shape or form equals x^-2. Where are you seeing that?

 

[mmm4444bot]

what law/property allows you to [rewrite the binomial x - 2 as the exponential] x^(-2) ?

There is no such property or law. Maybe your materials contain errors.

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As an aside, there is one Real number for x such that x - 2 = x^(-2)

x = (1/6)(M) + (8/3)(M)^(-1) + 2/3

where M = (172 + 12 sqrt[177])^(1/3)

Cheers :cool: