Question 541277
You could use logarithms to work this problem, but I'm not sure that you have gotten that far along yet. So let's look at another way. 
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Recall the power rule of exponents. For example think of:
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{{{(x^3)^4}}}
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You can raise {{{x^3}}} to the 4th power by multiplying the two exponents 4 and 3 to get 12. Another way of looking at it is that you get the 4th power of a quantity by multiplying that quantity by itself 4 times. So x-cubed multiplied by itself 4 times is:
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{{{x^3*x^3*x^3*x^3}}} and you find that result by adding the exponents to get:
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{{{x^3*x^3*x^3*x^3 = x^12}}}
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A word of caution. Since an even power is involved, the base can either be positive or negative and you will get the same positive result because both a positive value and the same negative value will give the same positive result when they are raised to an even power.
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That's just an example to show you that the power rule of multiplying exponents actually works. 
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The general form of the power rule is:
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{{{(x^a)^b = x^(a*b)}}}
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Now with that in mind, we can view {{{x^(4/3)}}} in a couple of ways. We can think of {{{4/3}}} as being the product of {{{4*(1/3)}}}. So, using the power rule we can rewrite the left side of the problem equation as:
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{{{(x^4)^(1/3)}}} or as {{{(x^(1/3))^4}}}
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either way will help us solve the problem. Recall that the exponent {{{1/3}}} means the cube root. 
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Now think about the right side of the equation problem. And note that 81 is equal to {{{3^4}}}
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So let's rewrite the entire equation as:
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{{{(x^(1/3))^4=3^4}}}
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Now if we take the 4th root of both sides we get:
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{{{x^(1/3) = 3}}}
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and since the 4th root is "even" not "odd", the result can be either positive or negative.
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And we can read this as "the cube root of x equals either +3 or -3." This can be inverted to: What are the numbers that if you take the cube root of them, the answer is either +3 or -3? And this can be further translated also to the answers: +3 cubed is what number and -3 cubed is what number? The answers are 
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{{{ x = 3^3}}} and this is {{{x = 27}}} or
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{{{ x = (-3)^3}}} and this is {{{x = -27}}}
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Now check by back-tracking. If we take the cube root of 27, we get both +3 and -3, and then we raise these two numbers to the 4th power and we get the same answer ... +81.
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Or we could take both +27 and -27 and raise them to the 4th power and take the cube root of that to see if we get +81. Replace either +27 or -27 with their equivalent values of 3*3*3 (which is {{{3^3}}}) or (-3)*(-3)*(-3) (which is {{{(-3)^3}}}). Raise these two quantities to the 4th power and you have the positive value {{{3^12}}}. And finally take the cube root of that positive quantity by dividing its exponent by 3 to get the positive value {{{3^4=81}}}.
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Either way it checks out. The answers to this problem are that x can be either +27 or -27.
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Hope that as you think your way through this problem you understand a little more about a way that you can work with rational exponents.
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