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, 5a*5)
3. 10^3/10^2 = 10 (10^3-2)
4. -x^2y/(-x^2)^2 y^2 = I couldn't figure this one out, so I would like some help in the explanation of this problem.
#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
The correct answer is 10.
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.
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