Math  /  Algebra

QuestionDivide. 30y76y2+12y6y230y76y2+12y6y2=\begin{array}{l} \frac{30 y^{7}-6 y^{2}+12 y}{6 y^{2}} \\ \frac{30 y^{7}-6 y^{2}+12 y}{6 y^{2}}= \end{array}

Studdy Solution

STEP 1

What is this asking? We're asked to simplify a big fraction with yys all over the place! Watch out! Don't forget to simplify *each* term in the numerator separately.
It's like dividing up a delicious pizza - everyone gets their own slice!

STEP 2

1. Separate the fraction
2. Simplify each term

STEP 3

We've got one big fraction, but we can make it easier to handle by splitting it into smaller pieces.
Think of it like separating different colored candies - much easier to count that way!
We can rewrite our expression like this:
30y76y2+12y6y2=30y76y26y26y2+12y6y2 \frac{30y^7 - 6y^2 + 12y}{6y^2} = \frac{30y^7}{6y^2} - \frac{6y^2}{6y^2} + \frac{12y}{6y^2}

STEP 4

Let's look at 30y76y2\frac{30y^7}{6y^2}.
We can divide the numbers and the variables separately. 3030 divided by 66 is **5**, and y7y^7 divided by y2y^2 is y72=y5y^{7-2} = \boldsymbol{y^5}.
So, 30y76y2\frac{30y^7}{6y^2} simplifies to 5y5\boldsymbol{5y^5}.

STEP 5

Now, let's tackle 6y26y2\frac{6y^2}{6y^2}.
Anything divided by itself is **1**!
So, 6y26y2=1\frac{6y^2}{6y^2} = \boldsymbol{1}.

STEP 6

Finally, we have 12y6y2\frac{12y}{6y^2}. 1212 divided by 66 gives us **2**.
And yy divided by y2y^2 is y12=y1y^{1-2} = y^{-1}, which is the same as 1y\frac{1}{y}.
So, 12y6y2\frac{12y}{6y^2} simplifies to 2y\frac{2}{y}, or 2y\boldsymbol{\frac{2}{\boldsymbol{y}}}.

STEP 7

Now, let's put all the pieces back together!
We have 5y55y^5, 11, and 2y\frac{2}{y}.
Our simplified expression is 5y51+2y\boldsymbol{5y^5 - 1 + \frac{2}{y}}.

STEP 8

Our final answer is 5y51+2y5y^5 - 1 + \frac{2}{y}.

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