SAT II Math II : Single-Variable Algebra

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

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Example Questions

Example Question #1 : Simplifying Expressions

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

Possible Answers:

\(\displaystyle 3x-y\)

\(\displaystyle 3xy\)

\(\displaystyle 3x + y\)

\(\displaystyle 5x+3y\)

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

Correct answer:

\(\displaystyle 3x + y\)

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:

\(\displaystyle 3x + y\)

Example Question #4 : Simplifying Expressions

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

Possible Answers:

\(\displaystyle 2x +3y +3\)

\(\displaystyle 2x +3y -3\)

\(\displaystyle 4x +y +3\)

\(\displaystyle -4x - y - 3\)

\(\displaystyle -3x^2 -2y^2 - 3\)

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:

\(\displaystyle -4x - y - 3\)

Example Question #5 : Simplifying Expressions

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

Possible Answers:

\(\displaystyle x\)

\(\displaystyle x^2\)

\(\displaystyle \sqrt{x^3}\)

\(\displaystyle \sqrt{x^4}\)

\(\displaystyle x^{\frac{1}{4}}\)

Correct answer:

\(\displaystyle x^2\)

Explanation:

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

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

From there, there's nothing left to simplify.

Example Question #51 : Single Variable Algebra

Simplify \(\displaystyle \sqrt{x^{11}}\).

Possible Answers:

\(\displaystyle 5\sqrt{x}\)

\(\displaystyle x^{11}\)

\(\displaystyle x^5 \cdot \sqrt{x}\)

\(\displaystyle (\sqrt{x})^5\)

\(\displaystyle \sqrt[5]{x}\)

Correct answer:

\(\displaystyle x^5 \cdot \sqrt{x}\)

Explanation:

To begin, let's rewrite the equation so the square root is a fraction in the exponent:

\(\displaystyle x^{\frac{11}{2}}\)

From here, we can simplify the exponent:

\(\displaystyle x^5 \cdot x^{\frac{1}{2}}\)

Now we change the exponent fraction back into a square root:

\(\displaystyle x^5 \cdot \sqrt{x}\)

Example Question #52 : Single Variable Algebra

Simplify \(\displaystyle (\sqrt{4x^2})\cdot (\sqrt{25x})^2\).

Possible Answers:

\(\displaystyle 50x^3\)

\(\displaystyle 100x^2\)

\(\displaystyle 50x^2\)

\(\displaystyle 25x +2x\)

\(\displaystyle 5 \sqrt{2x}\)

Correct answer:

\(\displaystyle 50x^2\)

Explanation:

For the first square root, each term inside has a natural solution.  We can take the square root of each term individually because they are multiplied, and then combine them again:

\(\displaystyle 2x \cdot (\sqrt{25x})^2\)

For the second square root, we remember that the square root and a square cancel each other out, and we're left with just the term inside:

\(\displaystyle 2x \cdot 25x\)

We finish by multiplying the terms together:

\(\displaystyle 50x^2\)

Example Question #53 : Single Variable Algebra

Simplify \(\displaystyle 2x^2 -5x(1 - 8x)\).

Possible Answers:

\(\displaystyle 2x^2-13x-1\)

\(\displaystyle 42x^2 -5x\)

\(\displaystyle 2x^2-13x\)

\(\displaystyle -38x^2 -5x\)

\(\displaystyle -11x^2\)

Correct answer:

\(\displaystyle 42x^2 -5x\)

Explanation:

We start by distributing the \(\displaystyle -5x\) term through the parentheses:

\(\displaystyle 2x^2 -5x + 40x^2\)

Now we combine like terms.  Remember, we can't add variables if they have different exponent terms:

\(\displaystyle 42x^2 -5x\)

Example Question #54 : Single Variable Algebra

Simplify \(\displaystyle 2x^2+7x(-8x- 10)\).

Possible Answers:

\(\displaystyle -54x^2- 70x\)

\(\displaystyle 56x^2 - 70x\)

\(\displaystyle -124x^3\)

\(\displaystyle 2x^2 -x-10\)

\(\displaystyle 2x^2 -14x\)

Correct answer:

\(\displaystyle -54x^2- 70x\)

Explanation:

Start by distributing the \(\displaystyle 7x\) term:

\(\displaystyle 2x^2-56x^2- 70x\)

Now combine like terms.  Remember, if a variable has a different exponent, you can't add them:

\(\displaystyle -54x^2- 70x\)

Example Question #55 : Single Variable Algebra

Simplify \(\displaystyle -10x(9x+ 4) - 5x^2\).

Possible Answers:

\(\displaystyle -95x^2- 40x\)

\(\displaystyle 95x^2+40x\)

\(\displaystyle 85x^2 +40x\)

\(\displaystyle -135x^3\)

\(\displaystyle -90x^2 -45x\)

Correct answer:

\(\displaystyle -95x^2- 40x\)

Explanation:

Start by distributing the \(\displaystyle -10x\) term:

\(\displaystyle -90x^2- 40x - 5x^2\)

Now collect like terms.  Remember, you can't add or subtract variables that have different exponents:

\(\displaystyle -95x^2- 40x\)

Example Question #211 : Sat Subject Test In Math Ii

Simplify \(\displaystyle -x^2 -6x(8 + 6x)\).

Possible Answers:

\(\displaystyle -x^2 -36x -48\)

\(\displaystyle 35x^2+48x\)

\(\displaystyle - 37x^2-48x\)

\(\displaystyle -75x^3\)

\(\displaystyle (-x-8)(x+6)\)

Correct answer:

\(\displaystyle - 37x^2-48x\)

Explanation:

Start by distributing the \(\displaystyle -6x\) term:

\(\displaystyle -x^2 -48x - 36x^2\)

Now combine like terms.  Remember, you can't add or subtract variables with different exponents:

\(\displaystyle - 37x^2-48x\)

Example Question #212 : Sat Subject Test In Math Ii

Simplify:  \(\displaystyle -\frac{2}{3}x^2 + (x)(x)(\frac{1}{2})\)

Possible Answers:

\(\displaystyle \frac{1}{6}x^2\)

\(\displaystyle -\frac{1}{6}x^2\)

\(\displaystyle 2x^2\)

\(\displaystyle -2x^2\)

\(\displaystyle -\frac{7}{6}x^2\)

Correct answer:

\(\displaystyle -\frac{1}{6}x^2\)

Explanation:

Multiply the right terms.

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

Convert to common denominators.

\(\displaystyle -\frac{2}{3}x^2 + \frac{1}{2}x^2 = -\frac{4}{6}x^2 + \frac{3}{6}x^2 = -\frac{1}{6}x^2\)

The answer is:  \(\displaystyle -\frac{1}{6}x^2\)

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