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ISEE Upper Level Math : Variables and Exponents

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Example Questions

Example Question #1 : How To Add Exponential Variables

Rewrite the polynomial in standard form:

$9x^{4} - 3x ^{2} + 8x^{5} -2x + 7 - 4x^{3}$

Possible Answers:

$8x^{5} + 9x^{4}- 4x^{3} - 3x ^{2} -2x + 7$

$-2x - 3x ^{2}- 4x^{3}+ 7+ 8x^{5} + 9x^{4}$

$9x^{4} + 8x^{5} + 7- 4x^{3} - 3x ^{2} -2x$

$7-2x- 3x ^{2}- 4x^{3}+ 9x^{4} + 8x^{5}$

$9x^{4} + 8x^{5} + 7-2x - 3x ^{2} - 4x^{3}$

Correct answer:

$8x^{5} + 9x^{4}- 4x^{3} - 3x ^{2} -2x + 7$

Explanation:

The degree of a term of a polynomial with one variable is the exponent of that variable. The terms of a polynomial in standard form are written in descending order of degree. Therefore, we rearrange the terms by their exponent, from 5 down to 0, noting that we can rewrite the and constant terms with exponents 1 and 0, respectively:

$9x^{4} - 3x ^{2} + 8x^{5} -2x + 7 - 4x^{3}$

$= 9x^{4} - 3x ^{2} + 8x^{5} -2x ^{1} + 7x ^{0} - 4x^{3}$

$=8x^{5} + 9x^{4}- 4x^{3} - 3x ^{2} -2x ^{1}+ 7x^{0}$

$=8x^{5} + 9x^{4}- 4x^{3} - 3x ^{2} -2x+ 7$

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Example Question #1 : How To Add Exponential Variables

Simplify:

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

Possible Answers:

$\frac{4}{x^{3}y^{3}}$

$\frac{4}{x^{2}y^{2}}$

$\frac{4}{x^{4}y^{4}}$

$\frac{4}{xy}$

Correct answer:

$\frac{4}{x^{2}y^{2}}$

Explanation:

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

$=\frac{(x^{4}+2x^{2}y^{2}+y^{4}) - (x^{4}-2x^{2}y^{2}+y^{4}) }{x^{4}y^{4}}$

$=\frac{(x^{4}-x^{4}+2x^{2}y^{2}+2x^{2}y^{2}+y^{4}-y^{4}) }{x^{4}y^{4}}$

$=\frac{4x^{2}y^{2}}{x^{4}y^{4}}$

$=\frac{4}{x^{2}y^{2}}$

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Example Question #2 : How To Add Exponential Variables

Assume that $x,y \neq 0$ . Simplify:

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

Possible Answers:

$\frac{2x^{4}-2y^{4} }{x^{4}y^{4}}$

$\frac{4}{x^{2}y^{2}}$

$\frac{2x^{4}+2y^{4} }{x^{4}y^{4}}$

$2x^{4}+2y^{4}$

$2x^{4}-2y^{4}$

Correct answer:

$\frac{2x^{4}+2y^{4} }{x^{4}y^{4}}$

Explanation:

$\frac{(x^{2}+y^{2})^{2} + (x^{2} + y^{2})^{2}}{x^{4}y^{4}}$

$=\frac{(x^{4}+2x^{2}y^{2}+y^{4}) + (x^{4}-2x^{2}y^{2}+y^{4}) }{x^{4}y^{4}}$

$=\frac{x^{4}+x^{4}+2x^{2}y^{2}-2x^{2}y^{2}+y^{4}+y^{4} }{x^{4}y^{4}}$

$=\frac{2x^{4}+2y^{4} }{x^{4}y^{4}}$

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Example Question #3 : How To Add Exponential Variables

If $x+y\neq 0$ , simplify:

$\frac{(x-y)^2+4xy}{x+y}$

Possible Answers:

x+y

x-y

Correct answer:

x+y

Explanation:

$\frac{(x-y)^2+4xy}{x+y}=\frac{(x^2-2xy+y^2)+4xy}{x+y}$

$=\frac{(x^2-2xy+4xy+y^2)}{x+y}$

$=\frac{(x^2+2xy+y^2)}{x+y}$

$=\frac{(x+y)^2}{x+y}=x+y$

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Example Question #3 : Variables And Exponents

If $x\neq 0$ , simplify:

$\frac{(x+\frac{1}{x})^3+(x-\frac{1}{x})^3}{\frac{2}{x}}$

Possible Answers:

x^4+3

x^2+3

x+3

Correct answer:

x^4+3

Explanation:

$\frac{(x+\frac{1}{x})^3+(x-\frac{1}{x})^3}{\frac{2}{x}}$

$=\frac{(x^3+3x^2\times \frac{1}{x}+3x\times \frac{1}{x^2}+\frac{1}{x^3})+(x^3-3x^2\times \frac{1}{x}+3x\times \frac{1}{x^2}-\frac{1}{x^3})}{\frac{2}{x}}$

$=(x^3+x^3)+(3x-3x)+(\frac{3}{x}+\frac{3}{x})+(\frac{1}{x^3}-\frac{1}{x^3})$

$=\frac{2x^3+\frac{6}{x}}{\frac{2}{x}}$

$=\frac{\frac{2x^4+6}{x}}{\frac{2}{x}}$

$=\frac{2x^4+6}{2}$

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

=x^4+3

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Example Question #2 : How To Add Exponential Variables

Define g(x) = 2x - 5

What is $g(x ^{2} + 3x)$ ?

Possible Answers:

$2 x ^{2} + 3x - 5$

$x ^{2} + 6x - 5$

$2 x ^{2} + 6x - 5$

$2 x ^{2} + 6x - 10$

Correct answer:

$2 x ^{2} + 6x - 5$

Explanation:

Substitute $x ^{2} + 3x$ for in the definition:

g(x) = 2x - 5

$g(x ^{2} + 3x ) = 2 \left ( x ^{2} + 3x \right ) - 5$

$g(x ^{2} + 3x ) = 2 \left ( x ^{2}\right ) + 2 \left ( 3x \right ) - 5$

$g(x ^{2} + 3x ) = 2 x ^{2} + 6x - 5$

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Example Question #3 : How To Add Exponential Variables

Add:

x^3+4x^2-2x^3+3x+2x^2+4

Possible Answers:

-3x^3+6x^2+3x+4

-x^3+6x^2+3x+4

3x^3+6x^2+3x+4

x^3+6x^2+3x+4

Correct answer:

-x^3+6x^2+3x+4

Explanation:

x^3+4x^2-2x^3+3x+2x^2+4

=(x^3-2x^3)+(4x^2+2x^2)+3x+4

=-x^3+6x^2+3x+4

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Example Question #4 : Variables And Exponents

Simplify:

(x+y)^2+2x^2-2y^2

Possible Answers:

x^2+2xy-y^2

3x^2+2xy-y^2

3x^2+2xy+y^2

3x^2-2xy-y^2

-3x^2+2xy-y^2

Correct answer:

3x^2+2xy-y^2

Explanation:

We can expand the first term using FOIL:

(x+y)^2=x^2+2xy+y^2

(x+y)^2+2x^2-2y^2=x^2+2xy+y^2+2x^2-2y^2

Reorder the expression to group like-terms together.

(x^2+2x^2)+2xy+(y^2-2y^2)

Simplify by combining like-terms.

3x^2+2xy-y^2

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Example Question #6 : Variables And Exponents

Simplify:

(2x-y)^2-(x-2y)^2

Possible Answers:

3x^2+3y^2-4xy

3x^2+3y^2

x^2+y^2

3x^2-3y^2

3x^2+3y^2+4xy

Correct answer:

3x^2-3y^2

Explanation:

Expand each term by using FOIL:

(2x-y)^2-(x-2y)^2 = [(2x)^2-(2) (2x)(y)+(y)^2]-[(x)^2+(2) (x)(2y)-(2y)^2]

=4x^2-4xy+y^2-x^2+4xy-4y^2

Rearrange to group like-terms together.

(4x^2-x^2)+(-4xy+4xy)+(y^2-4y^2)

Simplify by combining like-terms.

3x^2-3y^2

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Example Question #7 : Variables And Exponents

Simplify:

4x^3y-x^2y^2-3x^3y+2x^2y^2

Possible Answers:

-x^3y+x^2y^2

-x^3y-x^2y^2

x^3y-x^2y^2

x^3y+x^2y^2

The expression can not be simplified further

Correct answer:

x^3y+x^2y^2

Explanation:

Start by reordering the expression to group like-terms together.

4x^3y-x^2y^2-3x^3y+2x^2y^2

(4x^3y-3x^3y)+(-x^2y^2+2x^2y^2)

Combine like-terms to simplify.

(4x^3y-3x^3y)+(-x^2y^2+2x^2y^2)

x^3y+x^2y^2

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : Variables and Exponents

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : How To Add Exponential Variables

Example Question #1 : How To Add Exponential Variables

Example Question #2 : How To Add Exponential Variables

Example Question #3 : How To Add Exponential Variables

Example Question #3 : Variables And Exponents

Example Question #2 : How To Add Exponential Variables

Example Question #3 : How To Add Exponential Variables

Example Question #4 : Variables And Exponents

Example Question #6 : Variables And Exponents

Example Question #7 : Variables And Exponents

All ISEE Upper Level Math Resources

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