## anonymous one year ago ^3√(2^6)^x I know that the answer is 4^x but what are the steps to solve?

1. SolomonZelman

$$\LARGE \displaystyle \left(\sqrt[3]{2^6}\right)^x$$ like this?

2. anonymous

parentheses around only the 2^6

3. SolomonZelman

$$\LARGE \displaystyle\sqrt[3]{ \left(2^6\right)^x}$$

4. anonymous

yes like that

5. SolomonZelman

well, this is what you can do: $$\LARGE \displaystyle\sqrt[3]{ \left(2^6\right)^x}=\left(\sqrt[3]{ 2^6}\right)^x$$

6. SolomonZelman

then, deal with what is inside the parenthesis alone, and if the part inside the parenthesis is a 4, then the expression is 4$$^x$$.

7. SolomonZelman

more percisely. $$\LARGE \displaystyle\sqrt[3]{ \left(2^6\right)^x}=$$ $$\LARGE \displaystyle \left(\sqrt[3]{ 2^6}\right)^x=$$ $$\LARGE \displaystyle \left(2^{6/3}\right)^x=$$ $$\LARGE \displaystyle \left(2^{2}\right)^x=$$ $$\LARGE \displaystyle \left(4\right)^x.$$

8. SolomonZelman

if you have a question about the validity of the steps (i.e. what propery did I use for a particular step, or why doing something is valid and works, etc) or other questions as wel...

9. anonymous

Ok I understand it now... I just wasn't sure about about what to do with the 6 and the 3 but it makes sense now that I see it. Thank you so much!

10. SolomonZelman

Anytime..... But, what I did in the first step is not alwyas valid, although is in this case.

11. SolomonZelman

For example, if I say: $$\LARGE \sqrt{x^2}=\left( \sqrt{x} \right)^2$$ then I am faulty

12. SolomonZelman

Why? because the left side is the absolute value of x (i.e. |x| ) And it is continous for all values of x over the interval (-∞,+∞) And the right side is a line y=x for all values of x, where x≥0 (but undefined for x<0)

13. SolomonZelman

But, you can always use the following property: $$\LARGE \sqrt[a]{x^b}=x^{b/a}$$

14. SolomonZelman

that is what I used to get from line 2 to line 3