Using the Rules of Exponents. Please show me how to simplify each expression.
27 -2/3
____
27 -1/3
I would reorder the entire fraction first, to make working with the fractions easier
-2/3+27
________
-1/3+27
Then we should multiply each side of the fraction by the denominator's conjugate.
-2/3+27 (1/3+27)
________________
-1/3+27 (1/3+27)
-2/3+27 (1/3+27) =6478/9
-1/3+27 (1/3+27) =6560/9
You now have a fraction divided by another fraction. you can reorder them by using the reciprocal of the denominator fraction, which allows you to now multiply the two fractions.
6478
____
9
_________
6560
____
9
turns into...
6478 9
____ x _____
9 6560
From here, you multiply across.
You now have:
58302 79
______ Which simplifies into: __
59040 80
79/80 is your final simplified answer. Sorry this is so long, but this is a tough problem.
Well, the spacing was kinda off on that as usual, but hopefully you get the idea.
To simplify the given expression using the rules of exponents, we can rewrite the denominator as a fraction with the same base raised to a positive exponent:
27^(-1/3) = 1 / (27^(1/3))
Now, let's simplify the numerator:
27^(-2/3) = (1 / 27^(1/3))^2
According to the rules of exponents, when we raise a power to another power, we multiply the exponents. Therefore, we have:
(1 / 27^(1/3))^2 = 1^2 / 27^((1/3) * 2)
Simplifying further:
1^2 = 1
(1/3) * 2 = 2/3
Substituting these values back into the expression:
1 / 27^(2/3)
So, the simplified expression is 1 / 27^(2/3).