Question 1

Evaluate the algebraic expression $$-x^2 - 5x$ for x = -4$.

Accepted Answer

$$-16 - 20 = -36$$

Question 2

Use the laws of exponents to simplify the expression $$\left( \dfrac{a^2$ b$$^{-3}$$}{a^3$ b^2$} \$$right)^{-2}$$ using only positive exponents. Assume all variables represent nonzero real numbers.

Accepted Answer

Question 3

Find the product $$(5x + 4)(5x - 4). If $$you use a special formula, state the general formula that was used.

Accepted Answer

$$25x^2 - 16$; Difference of Squares: (a+b)(a-b$$) = $$a^2$$ - $$b^2$$

Question 4

Multiply $$\left( \sqrt{10x} - 9 \right) \sqrt{2x}$$ and simplify.

Accepted Answer

$$\sqrt{20x^2$} - 9\sqrt{2x}$$

Question 5

Factor the polynomial $$2xy - 3x - 10y + 15 by $$grouping.

Accepted Answer

$$(2xy - 3x) - (10y - 15) = x(2y - 3) - 5(2y - 3) = (x - 5)(2y - 3)$$

Question 6

Simplify the following expression, assuming all denominators are nonzero: $$\dfrac{16x - 12}{35} \div \dfrac{10}{3 - 4x}.$$

Accepted Answer

$$\dfrac{(16x - 12)(3 - 4x)}{350}$$

Question 7

Simplify the complex rational expression $$\dfrac{\dfrac{4}{x^2$} + 1}{\dfrac{1}{x^2$} + \dfrac{1}{x}}.$$

Accepted Answer

$$\dfrac{4 + x^2$}{1 + x}$$

Question 8

How many liters of pure water should be mixed with a $$5-L $$solution of $$80\%$$ acid to produce a mixture that is $$70\%$$ water?

Accepted Answer

$$15 L$$

Question 9

A pharmaceutical company needs to dilute a $$12-L $$solution of $$60\%$$ alcohol to obtain a $$40\%$$ alcohol solution. How much water should be added?

Accepted Answer

$$8 L$$

Question 10

If $$f(x) = 2x^2$ - 3x + 1$$, what is $$f(-2)$$?

Accepted Answer

$$2(-2)^2 - 3(-2) + 1 = 8 + 6 + 1 = 15$

Question 11

Which of the following is the correct factorization of $$x^2 - 9x + 20$?

Accepted Answer

$$(x - 5)(x - 4)$$

Question 12

A biologist is studying the growth of a bacteria population modeled by $$P(t) = 500 \cdot 2^t$$, where $$t is in $$hours. What is the population after $$3$$ hours?

Accepted Answer

$$500 \cdot 2^3$ = 500 \cdot 8 = 4000$$

Question 13

Simplify the expression $$\dfrac{2x^3$ y^2$}{4x y^5$}$$ and write your answer with positive exponents only.

Accepted Answer

$$\dfrac{x^2$}{2y^3$}$$

Question 14

A chemist has a $$10-L $$solution that is $$30\%$$ salt. How many liters of pure water must be added to reduce the concentration to $$20\%$$ salt?

Accepted Answer

$$10 L$$

Question 15

Which of the following is equivalent to $$\sqrt{50x^3$}$$?

Accepted Answer

$$5x^{3/2}$$\sqrt{2}$$

Precalculus Exam Study Guide: Algebraic Expressions, Exponents, Factoring, and Mixture Problems

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