We want to simplify the given expressions.
Following the order of operations,
we begin by simplifying inside the parentheses.
For the first expression, we begin by simplifying
three x cubed divided by nine x.
Looking at the coefficients, notice 3/9 simplifies to 1/3.
We're simplifying out the common factor of three.
Three divided by three simplifies to one,
and nine divided by three simplifies to three.
So we have the square of 1/3,
and there we have x cubed divided by x,
or x cubed divided by x to the first.
Remember when dividing and the bases are the same,
we subtract the exponents shown here
in the quotient property of exponents.
X cubed divided by x to the first is equal to x,
raised the power of three minus one,
which gives us the square of 1/3 x squared.
Now, there's a couple ways of simplifying from here.
Everything inside the parentheses is being squared,
and therefore this is equal to the square of 1/3.
And then we have the square of x squared.
Because we have powers raised to powers,
we multiply the exponents, which gives us x
raised to the power of two times two.
The square of 1/3 is equal to to 1/3 times 1/3 which is 1/9.
We have 1/9 x to the fourth.
Another way to simplify this going back to this step,
will be just to expand because 1/3 x squared
is being squared, we have two factors of 1/3 x squared.
So this is equal to 1/3 x squared times 1/3 x squared
which also gives us 1/9 x to the fourth.
Also I want to mention this expression can be written
in a slightly different form.
1/9 x to the fourth is equivalent to 1/9 times
x to the fourth over one, which is equal
to x to the fourth divided by nine.
So 1/9 x to the fourth is equivalent
to x to the fourth divided by nine.
Looking at our second example,
simplifying inside the parentheses first,
negative 4/6 simplifies to negative 2/3.
We're simplifying out the common factor of two.
Negative four divided by two simplifies to negative two.
Six divided by two simplifies to three.
So we have the square of negative 2/3,
and then we have x to the fourth divided by x to the second
which is equal to x raised to the power of four minus two.
Then we have y squared divided by y squared
which simplifies to y raised to the power of two minus two.
Simplifying, we have the square of negative 2/3 x squared
and then y raised to the power of zero is equal to one.
And now again, there's two ways to simplify this.
One way is to square everything inside the parentheses.
And therefore this is equal to the square of negative 2/3
times the square of x squared which is equal
to x raised to the power of two times two.
The square of negative 2/3 is equal to positive 4/9.
This simplifies to positive 4/9 x to the fourth.
Another way to simplify from here
would be to expand and write this as negative 2/3
x squared times negative 2/3 x squared
which gives us the same result of 4/9 x to the fourth.
But again, I do want to mention, this can be written
in a slightly different form.
4/9 x to the fourth is equivalent to 4/9 times
x to the fourth over one which gives us
four x to the fourth divided by nine which is
an equivalent way to write the simplified expression.
I hope you found this helpful.