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Well, let's see what we've got here:
( 4⁻³ · 3⁴ · 4² ) / ( 3⁵ · 4⁻² ) .
The two simple rules of exponents that we're
going to use here are:
-- Two multiply two numbers with the same base,
add their exponents. Like (A³) · (A²) = A⁵ .
But the bases have to be the same.
-- A negative power just means it belongs in the
other section of a fraction.
A negative power on top means it belongs on the bottom.
A negative power on the bottom means it belongs on top.
Like A⁻² means 1/A² . And 1/B⁻³ means B³ .
That's all you need in order to clean up the big fraction
in the question. But in order to see where you can use
these rules, you need to re-arrange things first.
Original: ( 4⁻³ · 3⁴ · 4² ) / ( 3⁵ · 4⁻² )
Let's send the 4⁻³
to the bottom
where it belongs: ( -- 3⁴ · 4² ) / 4³ · ( 3⁵ · 4⁻² )
Not take that 4⁻² from
the bottom, and put it on
top, where it belongs: ( 4² · 3⁴ · 4² ) / 4³ · ( 3⁵ -- )
Multiply the 4²s on top: ( 4⁴ · 3⁴ ) / ( 4³ · 3⁵ )
Now let me break this up.
Not changing anything, just
writing it in a different way: ( 4⁴ / 4³ ) · ( 3⁴ / 3⁵ )
Look at the first fraction: 4⁴ / 4³ .
Divide top and bottom by 4³ , and it becomes just 4 .
Now look at the second fraction: 3⁴ / 3⁵ .
Divide top and bottom by 3⁴ , and it becomes just 1/3 .
So in the end, we're left with just 4 / 3 .
And THAT is exactly equal to the original big messy fraction
in the question. It has exactly the same numerical value, but
you'd never know it when you see it, because it's a lot simpler.
There are a lot of other ways we could have manipulated and
massaged the original fraction, but the steps are the same:
-- Multiply numbers with the same base, by adding the exponents.
-- Remember that a number with a negative exponent belongs
in the other section of the fraction, with a positive exponent.