Question 554487
Given to solve for x:
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{{{x^(3/4) = 125}}}
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One way to solve this is to make use of the power rule of exponents. This rule says:
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{{{(x^a)^b = x^(a*b)}}}
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Or in words ... if you have a base quantity with an exponent, and you raise it to a power, it is equal to the base quantity raised to product of the two exponents.
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So what would happen if we raised {{{x^(3/4)}}} to the {{{4/3}}} power. It would be equal to raising x to the exponent {{{(3/4)*(4/3)}}}. But in multiplying the two exponents you get {{{12/12 = 1}}}. So raising {{{x^(3/4)}}} to the {{{4/3}}} power, it just becomes {{{x^1}}} which is just x.
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However, if you raise one side of this equation to the {{{4/3}}} power, you must do the same to the other side to maintain the equality. In equation form this is now:
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{{{x = 125^(4/3)}}}
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The product rule means that we can interpret the exponent as the product of {{{4*(1/3)}}} The right side of this equation can be interpreted in two ways. Either you can raise 125 to the 4th power and then take the cube root (the exponent 1/3 means cube root) of that answer. Or you can find the cube root (from the 1/3) of 125 and then raise that result to the 4th power.
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It's easier to do the second method. The cube root of 125 is 5. Then raise that to the 4th power to get 5*5*5*5 = 625. That's the answer to this problem:
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{{{x = 625}}}
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Hope this helps you in your understanding of exponents.