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ISEE Upper Level Math : How to divide variables

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Example Questions

Example Question #221 : Algebraic Concepts

Divide:

$\frac{-4x^{4}-6x^{3}+8x^{2}}{2x^{2}}$ $\displaystyle \frac{-4x^{4}-6x^{3}+8x^{2}}{2x^{2}}$

Possible Answers:

$-2x^{2}-3x+8$ $\displaystyle -2x^{2}-3x+8$

$-6x^{2}-8x+10$ $\displaystyle -6x^{2}-8x+10$

$-2x^{2}-3x+4$ $\displaystyle -2x^{2}-3x+4$

$-2x^{3}-3x^{2}+4x$ $\displaystyle -2x^{3}-3x^{2}+4x$

$-2x-3x^{2}+4$ $\displaystyle -2x-3x^{2}+4$

Correct answer:

$-2x^{2}-3x+4$ $\displaystyle -2x^{2}-3x+4$

Explanation:

To divide this problem we start by simplify it. In this problem we can break the large fraction into three smaller fractions.

$\frac{-4x^{4}-6x^{3}+8x^{2}}{2x^{2}}=\frac{-4x^{4}}{2x^{2}}-\frac{6x^{3}}{2x^{2}}+\frac{8x^{2}}{2x^{2}}$ $\displaystyle \frac{-4x^{4}-6x^{3}+8x^{2}}{2x^{2}}=\frac{-4x^{4}}{2x^{2}}-\frac{6x^{3}}{2x^{2}}+\frac{8x^{2}}{2x^{2}}$

From here we can factor out a 2x^2 $\displaystyle 2x^2$ from the numerator and denominator of each fraction.

$\frac{2x^2(-2x^2)}{2x^2}- \frac{2x^2(3x)}{2x^2}+ \frac{2x^2(4)}{2x^2}$ $\displaystyle \frac{2x^2(-2x^2)}{2x^2}- \frac{2x^2(3x)}{2x^2}+ \frac{2x^2(4)}{2x^2}$

Now we can cancel the 2x^2 $\displaystyle 2x^2$ terms and are left with the following result:

$-2x^{2}-3x+4$ $\displaystyle -2x^{2}-3x+4$

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Example Question #11 : How To Divide Variables

Simplify the following expression:

$\frac{343w^7}{49w^3}$ $\displaystyle \frac{343w^7}{49w^3}$

Possible Answers:

$\frac{49w^7}{w^3}$ $\displaystyle \frac{49w^7}{w^3}$

7w^4 $\displaystyle 7w^4$

343w^4 $\displaystyle 343w^4$

$\frac{343w^4}{49}$ $\displaystyle \frac{343w^4}{49}$

Correct answer:

7w^4 $\displaystyle 7w^4$

Explanation:

Simplify the following expression:

$\frac{343w^7}{49w^3}$ $\displaystyle \frac{343w^7}{49w^3}$

Let's begin by simplifying the coefficients

$\frac{343w^7}{49w^3}=\frac{49*7w^7}{49w^3}=\frac{7w^7}{w^3}$ $\displaystyle \frac{343w^7}{49w^3}=\frac{49*7w^7}{49w^3}=\frac{7w^7}{w^3}$

Next, complete the question by subtracting the exponents:

$\frac{7w^7}{w^3}=7w^{7-3}=7w^4$ $\displaystyle \frac{7w^7}{w^3}=7w^{7-3}=7w^4$

So, our answer is:

7w^4 $\displaystyle 7w^4$

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Example Question #221 : Algebraic Concepts

Simplify the following expression:

$\frac{49x^3y^5b^{12}}{7x^2y^6b^{8}}$ $\displaystyle \frac{49x^3y^5b^{12}}{7x^2y^6b^{8}}$

Possible Answers:

$\frac{7x^2b^4}{y}$ $\displaystyle \frac{7x^2b^4}{y}$

7xb^4y $\displaystyle 7xb^4y$

$\frac{49xb^4}{7y}$ $\displaystyle \frac{49xb^4}{7y}$

$\frac{7xb^4}{y}$ $\displaystyle \frac{7xb^4}{y}$

Correct answer:

$\frac{7xb^4}{y}$ $\displaystyle \frac{7xb^4}{y}$

Explanation:

Simplify the following expression:

$\frac{49x^3y^5b^{12}}{7x^2y^6b^{8}}$ $\displaystyle \frac{49x^3y^5b^{12}}{7x^2y^6b^{8}}$

Let's begin by simplifying the coefficients:

We have a 49 on top and a 7 on bottom. We can treat this just like a regular fraction:

$\frac{49}{7}=7$ $\displaystyle \frac{49}{7}=7$

Next, to deal with our variables, all we need to do is subtract the bottom exponent from the top exponent. If we get a negative number, we just put that number on the bottom.

$x^{3-2}=x^1=x$ $\displaystyle x^{3-2}=x^1=x$

$y^{5-6}=y^{-1}=\frac{1}{y}$ $\displaystyle y^{5-6}=y^{-1}=\frac{1}{y}$

$b^{12-8}=b^4$ $\displaystyle b^{12-8}=b^4$

So our answer will look like this:

$\frac{7xb^4}{y}$ $\displaystyle \frac{7xb^4}{y}$

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : How to divide variables

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #221 : Algebraic Concepts

Example Question #11 : How To Divide Variables

Example Question #221 : Algebraic Concepts

All ISEE Upper Level Math Resources

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