Simplify completely?

Question

1. (6x^-2y^3)^-3

2. (3x)^-3y^4 / -x^2y^-6

number two is a division problem.

appreciate any help! thank you! :]

Answer 1

1. Assuming you meant [(6xˆ-2)(yˆ3)]ˆ-3, then

Step 1: Separate the terms, each with the common exponent.

[(6xˆ-2)(yˆ3)]ˆ-3

(6xˆ-2)ˆ-3 • (yˆ3)ˆ-3

Step 2: Remember that an exponent raised to another exponent means that you multiply the exponents. So...

(6xˆ-2)ˆ-3 • (yˆ3)ˆ-3

[(6ˆ-3) • xˆ(-2 • -3)] • [yˆ(3 • -3)]

[216ˆ(-1) • xˆ6] • yˆ-9 (<==ANSWER!!)

Or you can rewrite it by placing all your negative exponents as the denominators in a fraction, making those exponents positive:

(xˆ6) / (216yˆ9) (<==ANSWER!!)

2. Assuming you meant, [(3x)ˆ-3 • (yˆ4)] / [(-xˆ2) • (yˆ-6)], then.

Step 1: Separate the fraction into two: having one fraction with the "x"s and the other with the "y"s.

[(3x)ˆ-3 • (yˆ4)] / [(-xˆ2) • (yˆ-6)]

[(3x)ˆ-3 / (-xˆ2)] • [(yˆ4) / (yˆ-6)]

Step 2: Expand the "(3x)ˆ-3".

[(3x)ˆ-3 / (-xˆ2)] • [(yˆ4) / (yˆ-6)]

(note: remember that xˆ-1 = 1/x, so that 3ˆ-3 = 27ˆ-1 = 1/27)

[(1/27)(xˆ-3) / (-xˆ2)] • [(yˆ4) / (yˆ-6)]

Step 3: Subtract the exponents that have the same base on the top and on the bottom. The sign (negative or positive) of the remaining exponent will determine whether the remaining expression will be at the numerator or the denominator of the fraction.

xˆ(-3 - 2) = xˆ-5 (<==negative exponent means denominator)

yˆ(4 - (-6)) = yˆ10 (<== positive exponent means numerator)

[(1/27)(xˆ-3) / (-xˆ2)] • [(yˆ4) / (yˆ-6)]

(yˆ10) / (27xˆ5) (<==ANSWER!!)

Hope I helped!!