Property of Exponents?

Question

1. (2^-1 cd^-4) (8c^-3 d^4)

originally i thought that the answer was 1/16c^2 but i'm not so sure anymore ... help! and explain please :)

2.) -12x^a+1 / 4x^2-a

no clue what to do form here?

Answer 1

1. 4 / c^2 if multiplying; c^4 / 16d^8 if dividing

2. -3 * x^(2a - 1)

When multiplying identical values (c * c, d * d, but not 2 * c) which are raised to powers, you add the exponents. c * c, for instance, is really c^1 * c^1 so you get c^(1 + 1) = c^2.

So, in the problems above, one way forward is to convert to multiplying everything by remembering how you do division by a fraction (like 1/2 divided by 1/3:  you "invert" the 1/3 and multiply) and "invert" the divisor portions of the problems. Which, for exponents means moving them above the division bar to multiply by them, but reversing their exponents' signs:  so c^2 / c^5 becomes c^2 * c^(-5) and d^2 / d^(-3) becomes d^2 * d^3. Then you do the adding exponents thing:  c^2 * c^(-5) = c^(2 + [-5]) = c^(-3).

Another way forward would be to realize that if multiplying means adding exponents, then dividing must mean subtracting the divisor's exponent from the denominator's:  c^2 / c^(-5) = c^(2 - [-5]) = c^(2 + 5) = c^7.

For problem 1, I'm assuming you meant for the first term to be multiplied by the second. But I'll do it both ways since you have conflicting results in your answer.

{2^(-1) * c * d^(-4)} * {8 * c^(-3) * d^4)}....change 8 to 2^3 and group terms

[2^(-1) * 2^3] * [c^1 * c^(-3)] * [d^(-4) * d^4]....rewrite with exponents being added

2^(-1 + 3) * c^(1 + [-3]) * d^([-4] + 4)....do the additions

2^2 * c^(-2) * d^0....d^0 = 1

2^2 * c^(-2)....calculate the 2^2; rewrite as division so "c" has a positive exponent

4 / c^2

{2^(-1) * c * d^(-4)} / {8 * c^(-3) * d^4)}....change 8 to 2^3

{2^(-1) * c * d^(-4)} / {2^3 * c^(-3) * d^4)}....group terms

[2^(-1) / 2^3] * [c^1 / c^(-3)] * [d^(-4) / d^4....now convert to subtracting those exponents

2^([-1] - 3) * c^(1 - [-3]) * d^([-4] - 4....do those operations

2^(-4) * c^4 * d^(-8)....rewrite as division to have only positive exponents

c^4 / (2^4 * d^8)....calculate 2^4 (= 16)

c^4 / 16d^8

For the second one, we do about the same steps:

(-12 * x^[a+ 1]) / (4 * x^[2 - a])....group the terms

(-12 / 4) * (x^[a+ 1] / x^[2 - a])....calculate -12/4; rewrite the "x" term combining exponents

-3 * x^([a+ 1] - [2 - a])....rewrite to be rid of the brackets in the "x" powers

-3 * x^(a + 1 - 2 + a)....combine the exponent terms

-3 * x^(2a - 1)