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Worksheet 1.2 Algebra Pretest
Attempt the problems below to self-assess topics that need attention. Use paper and pen or pencil and number each item so you may compare with the answers later. Do
NOT use a calculator, graphing tool, or AI to assist you. Many of the topics may be reviewed in
Chapter 2 ,
Chapter 3 ,
Chapter 4 , and
Chapter 5 below. Return to this pretest after you feel you have made sufficient progress. Answers may be found at the end of this chapter.
1. Working with Real Numbers.
Evaluate each expression. Your final answer should be a single simplified real number.
\(\displaystyle \frac{3}{2}-\frac{2}{3}\)
\(\displaystyle \frac{3}{2}\cdot\frac{2}{3}\)
\(\displaystyle |3−6|+6|−3|\)
\(\displaystyle \frac{3−6}{-3\times 6}\)
2. Working with Exponents.
Evaluate each expression. Your final answer should be a single simplified real number.
\(\displaystyle \left(-\sqrt{2}\right)^2-\left(\sqrt{2}\right)^4+\left(\sqrt{2}\right)^0+\sqrt{\left(-2\right)^4}\)
\(\displaystyle \sqrt{(-3)^2+(-4)^2}\)
\(\displaystyle \left(\sqrt[3]{-8}\right)^{-1}\)
\(\displaystyle \left(\frac{4}{9}\right)^{3/2}\)
\(\displaystyle 27^{-2/3}\)
\(\displaystyle \frac{2^{12}}{2^{10}}\)
3. Using Laws of Exponents.
Simplify each expression into the form
\(x^n\) and write your answer in the blank. Assume
\(x\neq 0\text{.}\)
\(\displaystyle \frac{x^5}{x^3}\)
\(\displaystyle x^5\cdot x^3\)
\(\displaystyle \left(x^5 x^1\right)^3\)
4. Converting to Exponential Form.
Rewrite the expression so that each term is in the form
\(a x^n\text{.}\) Assume
\(m \gt 0\text{.}\)
\begin{equation*}
\frac{3}{4x^5} - \sqrt{x^5} + \frac{1}{\sqrt[5]{x^2}} + \frac{2}{5 x^{-2}}
\end{equation*}
5. Using Scientific Notation.
Perform the indicated operation and express the result both in scientific notation and in standard decimal form.
\(\displaystyle \left(1.5\times 10^3 \right)\times\left(4.0 \times 10^{-2}\right)\)
\(\displaystyle \frac{6.5\times 10^3}{2.0 \times 10^{-2}}\)
\(\displaystyle \frac{6.5 \times 10^{-2}}{2.0\times 10^3} \)
\(\displaystyle \left(1.5\times 10^3 \right)+\left(4.0 \times 10^{-2}\right)\)
6. Reducing Radicals.
Reduce the radical expression as much as possible. Assume that any variables appearing may possibly be negative real numbers.
\(\displaystyle \sqrt{16 x^3 y^6 z^2}\)
\(\displaystyle \sqrt[3]{16 x^5 y^6 z^3}\)
\(\displaystyle \sqrt[4]{16 x^5 y^6 z^4}\)
7. Operations with Polynomial Expressions.
Perform the indicated operation and simplify as much as possible by reducing and combining like terms.
Subtract:
\(\left(4x^4-2x^3-3x+1\right)-\left(x^6-x^5+x^4-x^3+x^2-x+1\right)\)
Multiply:
\(\left(2x-3\right)\left(3x+1\right)\)
Multiply:
\(\left(4x-2y+1\right)\left(1-x+y^2\right)\)
8. Operations with Rational Expressions.
Perform the indicated operation and simplify as much as possible by reducing and combining like terms.
Subtract:
\(\displaystyle \frac{x}{x-3} - \frac{2x}{x+1}\)
Multiply:
\(\displaystyle \frac{x}{x-3} \cdot \frac{x+1}{2x}\)
Divide:
\(\displaystyle \frac{x}{x-3} \div \frac{x+1}{x^2-9}\)
9. Factoring Trinomials.
Completely factor each expression.
\(\displaystyle x^2+2 x-15\)
\(\displaystyle x^2-25x\)
\(\displaystyle x^2-10 x+25\)
\(\displaystyle 2 x^2-5 x-3\)
\(\displaystyle 2 x^2+5 x-3\)
\(\displaystyle x^2+2 x-15\)
\(\displaystyle 9x^4-y^2\)
\(\displaystyle a^2-b^2\)
\(\displaystyle a^2+2ab+b^2\)
\(\displaystyle a^2-2ab+b^2\)
10. Finding the Domain.
Find the domain of each expression, reporting your result in interval notation.
\(\displaystyle \sqrt{4-2x}\)
\(\displaystyle \frac{1}{4-2x}\)
\(\displaystyle \frac{1}{\sqrt{4-2x}}\)
\(\displaystyle \sqrt{4}-\sqrt{2x}\)
\(\displaystyle \sqrt[3]{4-2x}\)
11. Solving Equations.
\(\displaystyle x = 12-2x\)
\(\displaystyle x + x^2 = 12\)
\(\displaystyle x =- 4x^2\)
\(\displaystyle x^2 = 9\)
\(\displaystyle (x-3)^2 - 9 = 0\)
\(\displaystyle x^3 = 4x\)
\(\displaystyle |4-3x| = 1\)
\(\displaystyle \frac{1}{4-3x} = -2\)
\(\displaystyle \sqrt{4-3x} = 2\)
\(\displaystyle \frac{1}{x+3}-\frac{1}{x-3}=3\)
