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(a ^2 *b^2 *c ^ -3) *^ -4 *a^4 *c^-3 and that whole thing is divided by 2a^-2 *b^3

I was wondering how that would be simplified. I can only have positive exponents in the end. Can you please walk me through it?

Rassoul D. | Native French SpeakerNative French Speaker
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From the fact that for m and n,integers, and a, real number, (am)n = amn, (ambn)p = ampbnp,

am * an = am+n,  am / an = am-n,

(a2b2c-3)-4 * a4c-3 = a-8b-8c12a4c-3 = a-4b-8c9

So,  [(a2b2c-3)-4 * a4c-3]/2a-2b3 = a-4b-8c9/2a-2b3 = a-2b-11c9/2 = c9/2a2b11

Nicole C. | Increase confidence in math and science, chemistry, algebra IIIncrease confidence in math and science,...
4.9 4.9 (167 lesson ratings) (167)
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I'm going to write this in a little differently because it is how I am interpreting your question, hopefully I am reading the same thing you trying to get answered.

(a2·b2·c-3) -4· a4·c-3 ÷  (2a-2· b3)

We need to know rules for exponents. Combine like exponents (the exponents for x's together, etc.)

an exponent raised to an exponent: multiply the exponents  (x2)4 = x8

an exponent multiplied to and exponent: add the exponents  x2·x4 = x6

an exponent divided by an exponent: sub the bottom from the top  x2/x4 = x-2

a positive exponent means it belongs on the top of a fraction

a negative exponent means it belongs on the bottom of a fraction

Lets break it into two pieces and put back together at end

#1:  (a2·b2·c-3) -4· a4·c-3

distribute the -4 to the ( )

a-8 b-8 c12 · a4c-3

add exponents (because all the letters are being multiplied together)

a-4 b-8 c9

put the exponents on top and bottom of a fraction based on + or - sign or exponent

c9/a4b8

we will come back to this in a second

#2

2a-2· b3

put the exponents on top and bottom of a fraction based on + or - sign or exponent

2b3/a2

#1 and #2 together

(c 9/a4b 8)÷(2b 3/a2)

to divide fractions we flip the bottom one (the reciprocal) and multiply the fractions together

(c 9/a4b 8)×(a 2/2b 3)

multiply across the top to get new numerator, and across bottom to get new denominator

a2c 9/2a4b8b 3

add, sub, multiply as needed to simplify further

a: 2-4 = -2

b: 8+3 = 11

c: 9-0 = 9

c 9/2a2b11

5.0 5.0 (44 lesson ratings) (44)
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(a ^2 *b^2 *c ^ -3) *^ -4 *a^4 *c^-3 divided by 2a^-2 *b^3

I'm not sure what the *^ is supposed to mean. I'm going to assume that your problem looks like this:

[(a2b2c-3)-4a4c-3]/2a-2b3

The only operations are exponents, multiplication, and division.
This means we can bring the denominator up to the numerator if we change the sign of the exponents:

(a2b2c-3)-4a4c-3*2a2b-3

We want to get rid of the parentheses to an exponent. Remember (AmBn)w = AmwBnw

a-8b-8c12a4c-32a2b-3

2a(-8+4+2)b(-8-3)c(12-3)

2a-2b-11c9

That could be the answer, but you said you only want positive exponents. So, now we just put the terms with the negative exponents in the denominator and change their sign.

2c9/a2b11

Good luck on your test tomorrow.