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.

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:

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

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

2. In general:

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

3. Apply this rule to your question.

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.

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?

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 $\displaystyle \left(x^{-2} \right)^{\frac12}$

2. If you re-arrange the given term

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

you can see that

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

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

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.