Math  /  Algebra

QuestionDivide. 3by2y4÷15b8y\frac{3 b y}{2 y^{4}} \div \frac{15 b}{8 y}

Studdy Solution

STEP 1

What is this asking? We're asked to divide one algebraic fraction by another, which is like multiplying by its reciprocal! Watch out! Don't forget to flip the second fraction before multiplying, and watch out for those exponents when simplifying!

STEP 2

1. Rewrite as multiplication
2. Multiply the fractions
3. Simplify the result

STEP 3

Alright, let's **rewrite** our division problem as a multiplication problem.
Remember, dividing by a fraction is the same as multiplying by its reciprocal.
So, we **flip** the second fraction and **change** the division sign to multiplication: 3by2y4÷15b8y=3by2y48y15b \frac{3by}{2y^4} \div \frac{15b}{8y} = \frac{3by}{2y^4} \cdot \frac{8y}{15b}

STEP 4

Now, we **multiply** the numerators together and the denominators together.
It's like making one big fraction party! 3by2y48y15b=3by8y2y415b \frac{3by}{2y^4} \cdot \frac{8y}{15b} = \frac{3by \cdot 8y}{2y^4 \cdot 15b}

STEP 5

Let's **rearrange** things a bit to make it easier to see what's going on: 3by8y2y415b=38byy215by4 \frac{3by \cdot 8y}{2y^4 \cdot 15b} = \frac{3 \cdot 8 \cdot b \cdot y \cdot y}{2 \cdot 15 \cdot b \cdot y^4}

STEP 6

We can **simplify** the numbers by multiplying them out: 38byy215by4=24by230by4 \frac{3 \cdot 8 \cdot b \cdot y \cdot y}{2 \cdot 15 \cdot b \cdot y^4} = \frac{24by^2}{30by^4}

STEP 7

Time to **simplify**!
We can divide both the numerator and denominator by 6\boldsymbol{6}, their greatest common factor: 24by230by4=4by25by4 \frac{24by^2}{30by^4} = \frac{4by^2}{5by^4}

STEP 8

Now, let's **tackle** those variables.
We have bb in both the numerator and denominator, so we can divide both by bb to get 11: 4by25by4=41y251y4=4y25y4 \frac{4by^2}{5by^4} = \frac{4 \cdot 1 \cdot y^2}{5 \cdot 1 \cdot y^4} = \frac{4y^2}{5y^4}

STEP 9

We have y2y^2 in the numerator and y4y^4 in the denominator.
Remember, y4y^4 is the same as y2y2y^2 \cdot y^2.
So we can divide both the numerator and denominator by y2y^2: 4y25y4=4y25y2y2=45y2 \frac{4y^2}{5y^4} = \frac{4y^2}{5y^2y^2} = \frac{4}{5y^2}

STEP 10

Our **final simplified answer** is: 45y2 \frac{4}{5y^2}

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