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ISEE Upper Level Math : Variables and Exponents

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

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All ISEE Upper Level Math Resources

6 Diagnostic Tests 244 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : How To Subtract Exponential Variables

If $x\neq 0$ , simplify:

$\frac{(x^{2}+3)^2-(x^2-3)^2}{6x^2}$

Possible Answers:

2xy

$\frac{2}{x}$

Correct answer:

Explanation:

$\frac{(x^{2}+3)^2-(x^2-3)^2}{6x^2}=\frac{(x^4+6x^2+9)-(x^4-6x^2+9)}{6x^2}$

$=\frac{x^4-x^4+6x^2+6x^2+9-9}{6x^2}$

$=\frac{12x^2}{6x^2}=2$

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Example Question #31 : Variables And Exponents

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

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

Possible Answers:

2xy

Correct answer:

Explanation:

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

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

$=\frac{2xy}{xy}=2$

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Example Question #8 : How To Subtract Exponential Variables

If $x\neq 0$ , simplify:

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

Possible Answers:

$\frac{16}{x}$

$\frac{4}{x}$

Correct answer:

Explanation:

$(x+\frac{4}{x})^2-(x-\frac{4}{x})^2=(x^{2}+2\times x\times \frac{4}{x}+\frac{16}{x^2})-(x^{2}-2\times x\times \frac{4}{x}+\frac{16}{x^2})$

$=(x^{2}+8+\frac{16}{x^2})-(x^{2}-8+\frac{16}{x^2})$

$=x^2-x^2+8+8+\frac{16}{x^2}-\frac{16}{x^2}$

=8+8=16

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Example Question #31 : Variables And Exponents

Simplify the following expression:

12x^6-9x^6

Possible Answers:

3x^6

21x^6

2x^6

Correct answer:

3x^6

Explanation:

Simplify the following expression:

12x^6-9x^6

We can treat this basically as a regular subtraction problem.

We want to subtract our coefficients, while leaving the exponents the same.

12x^6-9x^6=3x^6

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Example Question #32 : Variables And Exponents

Simplify the following:

$(z^{3} x^{2} y)(x^{3} y^{4})$

Possible Answers:

$z^{3}x^{5}y^{5}$

$z^{3}x^{6}y^{4}$

$z^{3}xy^{3}$

None of the other answers

$(zxy)^{13}$

Correct answer:

$z^{3}x^{5}y^{5}$

Explanation:

To multiply variables with exponents, add the exponents. With multiple variables, simply add the exponents for each different variable.

Simplified:

$(z^{3})(x^{2+3})(y^{1+4})$

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Example Question #31 : Variables And Exponents

Simplify the following:

$(3py^{2})(4p^{4})$

Possible Answers:

$12(py)^{8}$

None of the other answers

$7p^{5}y^{2}$

$7py^{8}$

$12p^{5}y^{2}$

Correct answer:

$12p^{5}y^{2}$

Explanation:

To multiply variables with exponents, add the exponents. When there are constants mixed in, multiply the constants separately and put back in the final result:

$(3py^{2})(4p^{4})$

$3\times4\times p^{1+4}\times y^{2}$

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Example Question #34 : Variables And Exponents

Simplify the following:

$z^{2}\times z^{5}$

Possible Answers:

$z^{\frac{2}{5}}$

$z^{7}$

$z^{3}$

$z^{10}$

$z^{\frac{5}{2}}$

Correct answer:

$z^{7}$

Explanation:

To multiply variables with exponents, add the exponents. So,

$z^{2}\times z^{5}=z^{7}$

A longer way would be to write out all the multiplies the exponent tells us to do. This is a little clearer on why adding the exponents works but takes longer and isn't necessary once you understand the process.

$z^{2}\times z^{5}= zzzzzzz= z^{7}$

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Example Question #36 : Variables And Exponents

Factor completely:

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

Possible Answers:

$\left(x ^{2}- 2 \right ) \left (x + 2 \right )$

$x\left (x - 4 \right )\left (x - 1 \right )$

$x\left (x - 2 \right )\left (x - 2 \right )$

$x\left (x^{2} - 3x+ 4 \right )$

$\left(x ^{2}- 2 \right ) \left (x - 2 \right )$

Correct answer:

$x\left (x^{2} - 3x+ 4 \right )$

Explanation:

is the greatest common factor of each term, so distribute it out:

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

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

$= x \left (x^{2}- 3x + 4 \right )$

We try to factor $x^{2}- 3x + 4$ by finding two integers with product 4 and sum . However, both of our possible factor pairs fail, since -2 + (-2) = -4 and -1+ (-4) = -5 .

$x \left (x^{2}- 3x + 4 \right )$ is the complete factorization.

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Example Question #1 : How To Multiply Exponential Variables

Multiply:

$\left ( x -3y + 1\right ) \left ( x - 3y - 1\right )$

Possible Answers:

$x^{2} - 3xy +9y ^{2} - 1$

$x^{2} + 6xy -9y ^{2} - 1$

$x^{2} - 6xy +9y ^{2} - 1$

$x^{2} + 3xy -9y ^{2} - 1$

$x^{2} +9y ^{2} - 1$

Correct answer:

$x^{2} - 6xy +9y ^{2} - 1$

Explanation:

This can be achieved by using the pattern of difference of squares:

$\left ( x -3y + 1\right ) \left ( x - 3y - 1\right )$

$=\left [ \left ( x -3y \right ) + 1\right ] \left \left[( x - 3y \right ) - 1\right ]$

$= ( x -3y ) ^{2} - 1 ^{2}$

$= (x -3y) ^{2} - 1$

Applying the binomial square pattern:

$= x^{2} - 2x \left(3y \right ) +\left (3y \right ) ^{2} - 1$

$= x^{2} - 6xy +9y ^{2} - 1$

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Example Question #37 : Variables And Exponents

Factor completely:

$4 y^{2} - 100 y$

Possible Answers:

2 (y + 10) (4y - 5)

$4 (y - 5)^{2}$

$4y ^{2}(y - 25)$

4y (y - 25)

4 (y + 5) (y - 5)

Correct answer:

4y (y - 25)

Explanation:

The greatest common factor of the terms in $4 y^{2} - 100 y$ is , so factor that out:

$4 y^{2} - 100 y$

$= 4y\left ( \frac{4 y^{2}}{4y} - \frac{100 y}{4y} \right )$

$= 4y\left (y - 25 \right )$

Since all factors here are linear, this is the complete factorization.

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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 Subtract Exponential Variables

Example Question #31 : Variables And Exponents

Example Question #8 : How To Subtract Exponential Variables

Example Question #31 : Variables And Exponents

Example Question #32 : Variables And Exponents

Example Question #31 : Variables And Exponents

Example Question #34 : Variables And Exponents

Example Question #36 : Variables And Exponents

Example Question #1 : How To Multiply Exponential Variables

Example Question #37 : Variables And Exponents

All ISEE Upper Level Math Resources

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