SOLUTION: need help with steps to solving: (128x^21)^4/7

Question 606004: need help with steps to solving: (128x^21)^4/7
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
You are asked to simplify:
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The power rule of exponents says that you can raise each of the factors within the parentheses by the exponent that applies to the set of parentheses. So in this case you can raise the factor 128 and the factor x^21 by the exponent 4/7 as follows:
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Now let's look at the 128. It is being raised to the 4/7 power. The numerator 4 means take the 4th power, and the denominator 7 means take the 7th root. We can do that in one of two ways. We could raise 128 to the 4th power and then take the 7th root of that quantity. Or we could take the 7th root of 128 and then raise that to the 4th power. Either way, the result will be the same.
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It will be easier to use the second method, because it is fairly easy to find the 7th root of 128 by trial and error. It is 2 because if you multiply 2 by itself 7 times the answer is 128. So we know that the 7th root of 128 is 2 and then we raise 2 to the 4th power and the answer (2*2*2*2) is 16. So 128 to the 4/7th is 16.
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In the same way we need to raise x^21 to the 4/7th. We can either raise x^21 to the 4th power and then find the 7th root of that, or we can find the 7th root of x^21 and raise that to the 4th power. This time it doesn't make much difference which one of the two methods we use. We can raise x^21 to the 4th power by multiplying 21 times 4. The answer is x^84. Then we find the 7th root of that by dividing 84 by 7 and we get x^12. Or we could divide 21 by 7 first to get x^3 and then raise that to the 4th power by multiplying 3 * 4 to get x^12.
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Now we have the information to get the answer. We just multiply the two answers we got when we raised 128 to the 4/7th and when we raised x^21 to the 4/7th. The multiplication of 16 and x^12 is the answer. So we can say:
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Hope this helps you to understand the problem a little better.
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