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SAT II Math II : Simplifying Expressions

Study concepts, example questions & explanations for SAT II Math II

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2 Diagnostic Tests 130 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Simplifying Expressions

Decrease 8d - 35 $\displaystyle 8d - 35$ by 40%. Which of the following will this be equal to?

Possible Answers:

$\displaystyle 4.8 d - 21$

$\displaystyle 3.2d - 14$

4.8d-35 $\displaystyle 4.8d-35$

8d-21 $\displaystyle 8d-21$

3.2d-21 $\displaystyle 3.2d-21$

Correct answer:

$\displaystyle 4.8 d - 21$

Explanation:

A number decreased by 40% is equivalent to 100% of the number minus 40% of the number. This is taking 60% of the number, or, equivalently, multiplying it by 0.6.

Therefore, 8d - 35 $\displaystyle 8d - 35$ decreased by 40% is 0.6 times this, or

$0.6 \left (8d - 35 \right ) = 0.6 \cdot 8d - 0.6 \cdot 35 = 4.8 d - 21$ $\displaystyle 0.6 \left (8d - 35 \right ) = 0.6 \cdot 8d - 0.6 \cdot 35 = 4.8 d - 21$

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Example Question #2 : Simplifying Expressions

Simplify:

$\left ( 3x^{-3}\right )^{-3}$ $\displaystyle \left ( 3x^{-3}\right )^{-3}$

Possible Answers:

$27x^{9}$ $\displaystyle 27x^{9}$

$-27x^{9}$ $\displaystyle -27x^{9}$

$\frac{ x^{9} }{27}$ $\displaystyle \frac{ x^{9} }{27}$

$\frac{1}{27x^{9}}$ $\displaystyle \frac{1}{27x^{9}}$

$\frac{27}{ x^{9} }$ $\displaystyle \frac{27}{ x^{9} }$

Correct answer:

$\frac{ x^{9} }{27}$ $\displaystyle \frac{ x^{9} }{27}$

Explanation:

$\left ( 3x^{-3}\right )^{-3}$ $\displaystyle \left ( 3x^{-3}\right )^{-3}$

$= \left ( \frac{3}{x^{3}}\right )^{-3}$ $\displaystyle = \left ( \frac{3}{x^{3}}\right )^{-3}$

$= \left ( \frac{x^{3}}{3}\right )^{3}$ $\displaystyle = \left ( \frac{x^{3}}{3}\right )^{3}$

$= \frac{\left (x^{3} \right )^{3}}{3^{3}}$ $\displaystyle = \frac{\left (x^{3} \right )^{3}}{3^{3}}$

$= \frac{ x^{3 \cdot 3} }{27}$ $\displaystyle = \frac{ x^{3 \cdot 3} }{27}$

$= \frac{ x^{9} }{27}$ $\displaystyle = \frac{ x^{9} }{27}$

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Example Question #1 : Simplifying Expressions

Assume all variables assume positive values.

Simplify:

$\left ( \frac{y^{-3}}{x^{-2}} \right )^{2}$ $\displaystyle \left ( \frac{y^{-3}}{x^{-2}} \right )^{2}$

Possible Answers:

$\frac{x^{4}}{y^{6}}$ $\displaystyle \frac{x^{4}}{y^{6}}$

$\frac{y^{6}}{x^{4}}$ $\displaystyle \frac{y^{6}}{x^{4}}$

$\frac{x^{4}}{y^{9}}$ $\displaystyle \frac{x^{4}}{y^{9}}$

$\frac{1}{x^{4}y^{6}}$ $\displaystyle \frac{1}{x^{4}y^{6}}$

$\frac{y^{9}}{x^{4}}$ $\displaystyle \frac{y^{9}}{x^{4}}$

Correct answer:

$\frac{x^{4}}{y^{6}}$ $\displaystyle \frac{x^{4}}{y^{6}}$

Explanation:

$\left ( \frac{y^{-3}}{x^{-2}} \right )^{2}$ $\displaystyle \left ( \frac{y^{-3}}{x^{-2}} \right )^{2}$

$= \frac{y^{-3 \cdot 2}}{x^{-2\cdot 2}}$ $\displaystyle = \frac{y^{-3 \cdot 2}}{x^{-2\cdot 2}}$

$= \frac{y^{-6}}{x^{-4}}$ $\displaystyle = \frac{y^{-6}}{x^{-4}}$

$= y^{-6} \div x^{-4}$ $\displaystyle = y^{-6} \div x^{-4}$

$= \frac{1}{y^{6}} \div \frac{1}{ x^{4}}$ $\displaystyle = \frac{1}{y^{6}} \div \frac{1}{ x^{4}}$

$= \frac{1}{y^{6}} \cdot x^{4}$ $\displaystyle = \frac{1}{y^{6}} \cdot x^{4}$

$= \frac{x^{4}}{y^{6}}$ $\displaystyle = \frac{x^{4}}{y^{6}}$

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Example Question #1 : Simplifying Expressions

Assume all variables assume positive values. Simplify:

$6x^{0}+ 7y^{0}+ 8z^{0}$ $\displaystyle 6x^{0}+ 7y^{0}+ 8z^{0}$

Possible Answers:

The expression is already simplified.

21xyz $\displaystyle 21xyz$

The expression is undefined.

$\displaystyle 6x + 7y + 8z$

$\displaystyle 21$

Correct answer:

$\displaystyle 21$

Explanation:

Any nonzero expression raised to the power of 0 is equal to 1. Therefore,

$6x^{0}+ 7y^{0}+ 8z^{0}$ $\displaystyle 6x^{0}+ 7y^{0}+ 8z^{0}$

$= 6 \cdot 1 + 7 \cdot 1 + 8 \cdot 1$ $\displaystyle = 6 \cdot 1 + 7 \cdot 1 + 8 \cdot 1$

$\displaystyle = 6 + 7 + 8 = 21$

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Example Question #192 : Sat Subject Test In Math Ii

Simplify the following expression: $\displaystyle xyz(xy-xz)$

Possible Answers:

$\displaystyle x^2y^2z^2-x^2yz^2$

$\displaystyle 2xyz-2xz$

$\displaystyle x^2y^2z-x^2yz^2$

$\displaystyle x^2y^2z-x^2yz$

$\displaystyle 2x^2y^2z$

Correct answer:

$\displaystyle x^2y^2z-x^2yz^2$

Explanation:

Distribute the outer term through both terms in the parentheses.

$\displaystyle xyz(xy-xz) = xyz(xy) - xyz(xz)$

Multiply each term.

$\displaystyle x^2y^2z-x^2yz^2$

There are no like-terms.

The answer is: $\displaystyle x^2y^2z-x^2yz^2$

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Example Question #2 : Simplifying Expressions

Simplify the expression: $\displaystyle 6(2x-3)-2(-x-3)$

Possible Answers:

10x+9 $\displaystyle 10x+9$

12x-12 $\displaystyle 12x-12$

14x-24 $\displaystyle 14x-24$

10x-9 $\displaystyle 10x-9$

14x-12 $\displaystyle 14x-12$

Correct answer:

14x-12 $\displaystyle 14x-12$

Explanation:

Distribute the integers through the binomials.

$\displaystyle 12x-18 +2x+6$

Combine like-terms.

The answer is: 14x-12 $\displaystyle 14x-12$

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Example Question #3 : Simplifying Expressions

Simplify $\displaystyle x - 5 - (2 - x)$.

Possible Answers:

2x - 7 $\displaystyle 2x - 7$

$\displaystyle 0$

x^2 - 7 $\displaystyle x^2 - 7$

x+3 $\displaystyle x+3$

$\displaystyle -3$

Correct answer:

2x - 7 $\displaystyle 2x - 7$

Explanation:

We can start by distributing the negative sign in the parentheses term:

$\displaystyle x - 5 - 2 + x$

Now we can combine like terms. The constants go together, and the variables go together:

2x - 7 $\displaystyle 2x - 7$

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Example Question #3 : Simplifying Expressions

Simplify $\displaystyle -(x + y) + 4x + 2y$.

Possible Answers:

3x + y $\displaystyle 3x + y$

3x-y $\displaystyle 3x-y$

3xy $\displaystyle 3xy$

3(x+y) $\displaystyle 3(x+y)$

5x+3y $\displaystyle 5x+3y$

Correct answer:

3x + y $\displaystyle 3x + y$

Explanation:

First, we can distribute the negative sign through the parentheses term:

$\displaystyle -x - y + 4x + 2y$

Now we gather like terms. Remember, you can't gather different variables together. The $\displaystyle x$'s and $\displaystyle y$'s will still be separate terms:

3x + y $\displaystyle 3x + y$

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Example Question #4 : Simplifying Expressions

Simplify $\displaystyle -(x + 2y + 3) - 3x + y$.

Possible Answers:

$\displaystyle 2x +3y +3$

$\displaystyle 4x +y +3$

$\displaystyle 2x +3y -3$

$\displaystyle -4x - y - 3$

$\displaystyle -3x^2 -2y^2 - 3$

Correct answer:

$\displaystyle -4x - y - 3$

Explanation:

Start by distributing the negative sign through the parentheses term:

$\displaystyle -x - 2y - 3 - 3x + y$

Now combine like terms. Each variable can't be combined with different variables:

$\displaystyle -4x - y - 3$

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Example Question #5 : Simplifying Expressions

Simplify $(\sqrt{x^2})^2$ $\displaystyle (\sqrt{x^2})^2$

Possible Answers:

x^2 $\displaystyle x^2$

$\sqrt{x^3}$ $\displaystyle \sqrt{x^3}$

$x^{\frac{1}{4}}$ $\displaystyle x^{\frac{1}{4}}$

$\sqrt{x^4}$ $\displaystyle \sqrt{x^4}$

$\displaystyle x$

Correct answer:

x^2 $\displaystyle x^2$

Explanation:

A square root is the inverse of squaring a term, so they cancel each other out:

$(\sqrt{x^2})^2=x^2$ $\displaystyle (\sqrt{x^2})^2=x^2$

From there, there's nothing left to simplify.

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SAT II Math II : Simplifying Expressions

Study concepts, example questions & explanations for SAT II Math II

All SAT II Math II Resources

Example Questions

Example Question #1 : Simplifying Expressions

Example Question #2 : Simplifying Expressions

Example Question #1 : Simplifying Expressions

Example Question #1 : Simplifying Expressions

Example Question #192 : Sat Subject Test In Math Ii

Example Question #2 : Simplifying Expressions

Example Question #3 : Simplifying Expressions

Example Question #3 : Simplifying Expressions

Example Question #4 : Simplifying Expressions

Example Question #5 : Simplifying Expressions

All SAT II Math II Resources

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