math help?

Question

will someone please help me out?

silplify the following to the form of a^b where a and b are integers.

2048 times (1/2^5)^7 times (1/8)

what i did: 2048)(1/8)+256. simplified denominator. had: 256(1/32)^7 and mulitplied 256 by 1/32. and ended with 8^7, which is wrong.
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silplify the following to the form of a^b where a and b are integers.

(3125)^4 times (1/2^5)^7 times 5 times 125^12

what i did:multiplied whole numbers, got 1,953,125^16 . simplified denominator of fraction. and had 1,953,125^16 times (1/32)^7 multiplied and ended with 5.68^16, which isnt a whole number, i know.
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one more.
solve or simplify using laws of exponents.

(3)(5^x-5) all over (27)(5^x-3)

what i did: i divided 3 and 27 by 2. and subtracted exponents, and ended with 5^x-2 all over 9. apparently the answer is 1/225. please help?

Answer 1

In the first one, the power 7 only applies to the fraction, not to the factor in front.

However, you could use the fact that 2048 = 2^11 and 8 = 2^3 to "simplify" to

(2^11) * (1 / 2^35)*(1 / 2^3) = 2^11 * 2^-35 * 2^-3 = 2^-27

Law of exponents:

1 divided by a^b = a^-b

a^b times a^c = a^(b+c)

(a^b)^c = a^(bc)

Easy: 1^b = 1, whatever b is (even 1^0 = 1)

Answer 2

2048 is 2^11
((1/2)^5)^7 is 1/2^35
Multiplying this yields 2^12/2^35
So since you are dividing and have a like abse you subtract the exponents and get 2^12/2^35=2^-24
Now multiply by 1/8, which is equal to 1/2^3 and you again subtract the exponents and get 2^-27

*edited*

I had a couple errors, fixed now.