Tutorial 1: Pre-Calculus: Sets and Basic Algebra Involving Simplifying and Rationalizing Expressions

Quick Solutions to Pre-Calculus: Sets and Basic Algebra Involving Simplifying and Rationalizing Expressions

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Question 1.

Use the roster method to name each set:
1. \(A=\{x \mid x \mbox{ is an integer and } 1<= x <= 8 \}\)
2. \(B=\{y \mid y \mbox{ is an integer and } -5<= y <= 0 \}\)

Question 2.

Use the builder notation to name each set:
1. A=\(\{-1,0,1,2\}\)
2. B=\(\{10,11,12,13,\dots\}\)
3. C=\(\{10,20,30,40,\dots\}\)

Question 3.

Place \(\subset\), \(\supset\) or = in each blank to make a true sentence.
1. A=\(\{8,9\}\dots\dots\{7,11,9,8\}\)
2. B=\(\{5,4,3,2,1\}\dots\dots\{1,2,3\}\)
3. C=\(\{4,5,6\}\dots\dots\{6,4,5\}\)

Question 6.

Find each of the following intersections:
1. \(\{\frac{1}{2},1\} \cap \{-4,8\}\)
2. \(\{3,4,5,6,7,\dots\} \cap \{0,1,2,3,4\}\)
3. \(\{3,4\} \cap \emptyset \)
4. \(A \cap \emptyset \mbox{ for any set A}\)
5. \(\{x \mid x < 0\} \cap \{x \mid x < -1\}\)

Question 7.

Find each of the following unions.
1. \(\{\frac{1}{2},1\} \cup \{-4,8\}\)
2. \(\{3,4,5,6,7,\dots\} \cup \{0,1,2,3,4\}\)
3. \(\{1,2,3\} \cup \emptyset \)
4. \(A \cup \emptyset \mbox{ for any set A}\)
5. \(\{x \mid x < 0\} \cup \{x \mid x < -1\}\)
\(\{x \mbox{ an positive integer and even}\}\)

Question 10.

Name\(\{x \mid x \mbox{ is an odd integer\} in two other ways}\)

Question 12.

Find (n is a positive real number)
1. \( (-\infty,3)\cap [2,\infty)\)
2. \( (-\infty,3)\cup [3,\infty)\)
3. \( (-1,2)\cup [1,4)\)
4. \( (-1,2)\cap [1,4)\)
5. \( (3,3)\)
6. \( (n,n)\cap [-(n + 1),n + 1]\)
7. \( (-n,n)\cup [-(n + 1),n + 1]\)

Question 17.

Perform the indicated operations.
1. \(\frac{2}{5} \div \frac{9}{10}\)
2. \( (\frac{1}{8} - \frac{1}{9}) \div \frac{1}{12}\)
3. \( (2 \div \frac{2}{3}) - (\frac{2}{3} \div 2) \)

Question 18.

1. Evaluate each of the following given numbers:
(a) \(\sqrt[4]{24}. \sqrt[4]{54}\)
(b) \( (\frac{25}{64})^\frac{3}{2}\)
(c) \(3^\frac{2}{7}. 3^\frac{5}{7}\)
(d) \(1024^{-0.1}\)
2. Simplify the expression (assume that the letters denote any real numbers)
(a) \(\sqrt[3]{108} - \sqrt[3]{32}\)
(b) \(\sqrt{245} - \sqrt{125}\)
(c) \(\sqrt[5]{a^6b^7}\)
(d) \(\sqrt[3]{\sqrt{64x^6}}\)
(e) \(\sqrt[3]{a^2b}. \sqrt[3]{a^4b}\)
(f) \(\sqrt[3]{x^3y^6}\)
3. Rationalize the denominator
(a) \(\sqrt{\frac{x}{3y}}\)
(b) \(\frac{1}{\sqrt[3]{x}}\)
(c) \(\frac{1}{\sqrt[3]{x^4}}\)
4. Without using a calculator, determine which number is larger in each pair of numbers
(a) \(7^\frac{1}{4} \mbox{ and } 4^\frac{1}{4}\)
(b) \(\sqrt[3]{5}\) and \(\sqrt{3}\)