Math Help - more interested in the process?

Question

I am more interested in knowing how to achieve the answers w/o the benefit of a calculator) than simply what the answers are. (10 points to the first one that makes some sense to me)
Thanks in advance

1. simply: sqrt 49 + sqrt (-12)
2. simplify w/o neg. exponents (-2a^-3b^-7c^3)^-2
3. simply 27^(2/3)

Answer 1

I can help with a couple of them...but I am having trouble knowing what you are typing on #2.

1. sqrt 49 + sqrt (-12)
sqrt of 49 = 7
sqrt of (-12) must be broken down frist. 4 is a factor of 12 and a perfect square. So you will want the sqrt of 4 and (-3) then remember the sqrt (-1) is i. So broken down....

sqrt 49 + sqrt (-12)
7 + [sqrt 4 x sqrt 3 x sqrt (-1)]
7 + 2i sqrt 3
That is simplified.

#3
a fraction in an exponent means that you will use the top of the fraction as the exponent and the bottom as the "root"
so

27^(2/3) = cubed root of 27^2
or (cubed root of 27)^2
27 is a perect cube...so you can cube root it before squaring
= 3^2
= 9

There you go!

Answer 2

1. you can only add square roots if they are of the same value (ex. sqrt 2 + sqrt2 = 2 sqrt 2)

simplifying your square roots you want to find perfect square factors of the number.
49 is a perfect square of 7
-12 isn 't a perfect square but can be broken into -1*4*3 and 4 is a perfect square

sqrt 49 = 7
sqrt(-12) = i*2*sqrt3
answer: 7 + 2isqrt 3

2. powers on the outside go to each base on inside of () and powers to powers multiply together
(-2)^-2 a^6 b^14 c^-6

since the -2 and c are still to negative powers, they go to denominator since negative powers take the reciprocal of the base.
(a^6b^14)/(-2)^2c^6
(a^6b^14)/4c^6


3. the 2/3 power is equivalent to taking the cube root and then squaring the answerr

27^(2/3) = cuberoot(27)^2 = 3^2 = 9

Answer 3

1)
√49 + √-12
= √[7²] +√[2²×3×(-1)] ... Expressing as factors helps to see what is squared under the square root.
=7 + √[2²]×√[3]×√[(-1)]
= 7 + 2√3 i ... the square root of -1 is i.

2)
(-2a^-3b^-7c^3)^-2
= 1/(-2/a^3 × 1/b^7 × c^3)^2 ... Raised to a negative power just means writen as a fraction
= 1/(4/a^6 × 1/b^14 × c^6) .. when squaring we can just multiple the exponents by two.
= a^6b^14 / 4c^6 .. Then just neatening up the compound fractions.

3)
27^(2/3)
= (3^3)^(2/3)
= 3^(3*2/3)
= 3^2
= 9

Answer 4

1. 7+sqrt(3x4x-1)= 7+2sqrt(3)i

(a^b)^c=a^(bc) a^(-1)=1/a
2. a^6b^14/(4c^6)
3 (3^3)^(2/3)=3^2=9

Answer 5

the answer is 9