SOLUTION: Please help me How to solve this equation containing a rational exponent on the variable. x 4/3= 81

Question 541277: Please help me How to solve this equation containing a rational exponent on the variable.
x 4/3= 81

Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
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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You can raise 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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and you find that result by adding the exponents to get:
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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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Now with that in mind, we can view in a couple of ways. We can think of as being the product of . So, using the power rule we can rewrite the left side of the problem equation as:
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or as
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either way will help us solve the problem. Recall that the exponent 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
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So let's rewrite the entire equation as:
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Now if we take the 4th root of both sides we get:
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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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and this is or
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and this is
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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 ) or (-3)*(-3)*(-3) (which is ). Raise these two quantities to the 4th power and you have the positive value . And finally take the cube root of that positive quantity by dividing its exponent by 3 to get the positive value .
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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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