1-4. Are the following results correct? Simplify. Assume a, x , and y are not zero. 1. 8x/12 = 1x/3 (8*1,8*3) 2. 15a^3/25a = 3a^3/5a (3a^3*5,...

#3 correct.  Here are the explanations of the
problems.

1.  8x/12 can be simplified by
4. 

8x `-:` 4 = 2x 

12 `-:` 4
= 3

The correct answer is
2x/3.

 

2. 
15a^3 / 25a can be simplified by 5. 

15a^3 `-:` 5 =
3a^3

25a = 5a

3a^3 / 5a can be
simplified using exponential laws.  When dividing powers, you subtract the exponents. 
Note that 5a is actually 5a^1.

a^3 - a^1 =
a^2

A positive exponent places the variable in the
numerator.

The correct answer is 3a^2 /
5. 

3.  10^3 /
10^2

10^(3-2)

10^1

10

The
correct answer is 10.

4. 
-x^2y/(-x^2)^2y^2

For problems with multiple variables, I
recommend dealing with one variable at a time and then combining them at the
end.

You must first work the parentheses according to order
of operations.

(-x^2)^2

To
find the power of a power, you multiply the
exponents.

-x^(2*2) = -x^4

So
now (just dealing with the variable x), we have...

-x^2 /
-x^4

First of all, the negatives cancel, so now we
have...

x^2 / x^4 = x^(2-4) =
x^(-2)

Because the exponent is negative, x^2 will go in the
denominator in the final answer.  Now lets work with the variable
y.

y / 2y^2

To make things
easier, give y a coefficient and exponent of 1.

1y^1 /
2y^2

Set aside the constants (numbers).  The final fraction
will contain 1 / 2.

y^1 / y^2 = y^(1-2) =
y^-1

Because the exponent is negative, y^1 (or simply just
y) will go in the denominator of the final answer.

Now we
bring the answer together.

We have the fraction
1/2.

We know that x^2 is in the
denominator.

We know that y is in the
denominator.

The correct answer is 1 / (2x^2
* y).

Just as a review, here are
the basic exponentials laws:

n^a * n^b =
n^(a+b)

n^a `-:` n^b =
n^(a-b)

(n^a)^b = n^(a*b)

n^-a
= 1 / n^a