Question

Perform the operations and simplify.
((7x^3)^4/3)/
(7^1/3x^5)

Answer 1

Oh man....this is really hard to read. Is the numerator (7x^3) raised to the 4/3 or raised to the 4th, the whole thing divided by 3? And in the denominator, is 7 raised to the (1/(3x)^5), or is it 7^(1/3) x x^5?

Answer 2

raised to the 4/3

Answer 3

on the denominator the 7 is raised to the 1/3. and the x is raised to the 5th

Answer 4

Cool, that makes things easier. Alright, start with just the numerator:
\[(7x^3)^{4/3}\]When you have an exponent raised to an exponent (or things in parenthesis raised by an exponent, in this case 4/3), you can multiply the exponent across all the exponents inside the parenthesis.
7 is raised to the first power (an invisible 1), so multiplying that exponent by 4/3 gives you
7^(1 x 4/3) = 7^(4/3)

The x is raised to the third power, so multiplying that one gives you
x^(3 x 4/3) = x^4

So the numerator should look like this:
\[7^{4/3}x^4\]

Moving onto the denominator, there isn't any extra multiplying or simplifying left, so now you can start reducing the entire fraction.
Notice that you have a 7 raised to a power in both the numerator and denominator. When you divide like numbers raised to exponents, you subtract the exponent of the denominator (1/3) from the exponent in the numerator (4/3). So the 7s reduce like so:

7^(4/3 - 1/3)
7^(3/3)
7

That's a lot nicer to look at (and type). You repeat the same process with the exponents on the x values.

x^(4 - 5)
x^(-1)

Since your exponent is negative, the x stays in the denominator. Thus the fraction reduces to this:

7/x

Answer 5

awesome explanation, thank you!

Answer 6

No problem! Good luck!