What property does each of the following equations illustrate?

1. \(4+6=6+4\)

2. \(3(5-8)=3(5)-3(8)\)

3. \(2\cdot (3\cdot 5)=(2\cdot 3)\cdot 5\)

4. \(4\cdot 1=4\)

Write each of the following in symbols:

The sum of \(x\) and \(5\) is less than \(2\).
The product of \(3\) and \(x\) is \(21\).
The quotient of \(y\) and \(6\) is \(4\).
Twice the difference of \(b\) and \(7\) is greater than \(5\).
The difference of twice \(b\) and \(7\) is greater than \(5\).

Simplify: \(\; 5+3(2+4)\)

Simplify: \(40-20 \div 5+8\)

Simplify \((-2-3)(5-9)\)

Evaluate the following.

1. \(\lvert -3\rvert\)

2. \(-4-\lvert 4.5\rvert\)

3. \(\lvert -4+9\rvert +3\)

Simplify \(2-5(7-4)-6\)

Simplify \(2(4-7)^3+(-2-3)^2\)

Simplify each expression.

1. \(3^2+2(-3+7)\)

2. \(\dfrac{3^2+1}{4-7}\)

3. \(6+3(5-(-3^2+2))\)

Simplify: \(\dfrac{-8-8}{-5-3}\)

Evaluate the following expression on your calculator. Check the calculator output with a hand calculation.

1. \(4+10\div 5+2\)

2. \(\dfrac{4+10}{5+2}\)

Simplify: \(\dfrac{-5(-4)+2(-3)}{2(-1)-5}\)

Find the value of the expression \(3x+4y+5\) when \(x\) is \(6\) and \(y\) is \(7\)

Write \(\lvert 5\rvert\) without absolute value symbols.

Write \(\lvert -2\rvert\) without absolute value symbols

Write \(-\lvert -3\rvert\) without absolute value symbols

Simplify \(5(4x)\) by using the associative property.

Simplify \(\dfrac{1}{4}(4a)\) by using the associative property.

Apply the distributive property to \(\dfrac{1}{2}(3x+6)\) and then simplify the result.

Simplify: \(\,2(3y+4)+2\)

Apply the distributive property to \(a\left(1+\dfrac{1}{a}\right)\) and then simplify the result.

Apply the distributive property to \(3\left(\dfrac{1}{3}x+5\right)\) and then simplify the result.

Apply the distributive property to \(6\left(\dfrac{1}{3}x+\dfrac{1}{2}y\right)\) and then simplify the result.

Use the distributive property to combine similar terms for the expression \(3x+5x\).

Use the distributive property to combine similar terms for the expression \(3y+y\)

Simplify: \(7x+4+6x+3\)

Simplify: \(4+3(2y+5)+8y\)

Simplify: \(3(2y-1)+y\)

Simplify: \(8-3(4x-2)+5x\)

Simplify: \(5(2a+3)-(6a-4)\)

Let \(A=\{1, 3, 5\}\), \(B=\{0, 2, 4\}\), and \(C=\{1, 2, 3, \ldots\}\). Find \(A\cup B\).

Let \(A=\{1, 3, 5\}\), \(B=\{0, 2, 4\}\), and \(C=\{1, 2, 3, \ldots\}\). Find \(A\cap B\).

Let \(A=\{1, 3, 5\}\), \(B=\{0, 2, 4\}\), and \(C=\{1, 2, 3, \ldots\}\). Find \(A\cap C\).

Let \(A=\{1, 3, 5\}\), \(B=\{0, 2, 4\}\), and \(C=\{1, 2, 3, \ldots\}\). Find \(B\cup C\).

For the set \(\left\{-5, \,-3.5, \,0, \,\frac{3}{4}, \,\sqrt{3}, \,\sqrt{5}, \,9\right\}\), list:

1. whole numbers

2. integers

3. rational numbers

4. irrational numbers

5. real numbers

Write each expression using positive exponents.

1. \(4^{-2}\)

2. \(1.45^{-3}\)

3. \(\dfrac{x^{-4}}{y^5}\), \(y\neq 0\)

Simplify each expression and write it using positive exponents. Assume that variables represent nonzero real numbers.

1. \(\dfrac{16x^{10}y^4}{4x^{10}y^8}\)

2. \(\dfrac{\left(3x^{-2}\right)^3}{x^{-5}}\)

3. \(\left(\dfrac{54s^2t^{-3}}{6zt}\right)^{-2}\)

Express each of the following numbers in scientific notation.

1. \(328.5\)

2. \(4.69\)

3. \(0.00712\)

Write the following numbers in decimal form.

1. \(2.1\times 10^5\)

2. \(3.47\times 10^{-3}\)

In 2012, Japan had a population os \(1.27\times 10^8\) and a land area of \(3.64\times 10^5\) square kilometers. What is the population density of Japan, that is, the number of people per square kilometer of land? Express your answer in scientific notation.

Determine \(\sqrt[3]{64}\), \(\sqrt[4]{64}\), and \(\sqrt[5]{-32}\).

Rationalize the denominator of each expression.

1. \(\sqrt{\dfrac{18}{5}}\)

2. \(\dfrac{\sqrt[3]{10}}{\sqrt[3]{3}}\)

Simplify the following.

1. \(\sqrt{75}\)

2. \(\sqrt[3]{\dfrac{125}{108}}\)

3. \(\sqrt{x^5y^7}\), \(x,y>0\)

Rationalize the denominator.
\(\dfrac{5}{4-\sqrt{2}}\)

Simplify the following radical expressions. Assume \(x\geq0\).

1. \(\sqrt{48}+\sqrt{27}-\sqrt{12}\)

2. \(\left(3+\sqrt{x}\right)\left(4-2\sqrt{x}\right)\)

Evaluate the following.

1. \(27^\frac{1}{3}\)

2. \(-36^\frac{1}{2}\)

Evaluate the following expressions without using a calculator.

1. \((4)^\frac{1}{2}\)

2. \(\left(\dfrac{1}{3}\right)^{-2}\)

3. \((23.1)^0\)

Simplify and write positive exponents.

1. \((25)^\frac{1}{2}\)

2. \(\left(16s^\frac{4}{3}t^{-3}\right)^\frac{3}{2}\), \(s,t>0\)

3. \(\dfrac{\left(x^{-\frac{2}{3}}y^{-3}\right)^3}{y^{-\frac{1}{2}}x^2}\), \(x,y>0\)

Add or subtract each of the following.

1. \(\left(3x^3+2x^2-5x+7\right)+\left(x^3-x^2+5x-2\right)\)

2. \(\left(s^4+\dfrac{3}{4}s^2\right)-\left(s^4-s^2\right)\)

Multiply \(\left(-2x^2\right)\left(4x^7\right)\)

Multiply \((3x+2)(-2x-3)\)

Multiply \((-7x+4)(5x-1)\)

Multiply \((4y^3-3y+1)(y-2)\)

Factor \(x^3-x^2+2x-2\) by grouping.

Factor each of the following

1. \(x^2-2x-8\)

2. \(8x^3-10x^2-12x\)

For what value of \(x\) is the following rational expression defined? \[\frac{x+1}{(x-3)(x-5)}\]

Simplify: \(\dfrac{x^2-2x+1}{3x^2-4x+1}\)

Divide and simplify:
\(\dfrac{3x^2-5x-2}{x^2-4x+4}\div\dfrac{9x^2-1}{x+5}\)

Simplify \(\dfrac{\dfrac{x^2-4}{2x+1}}{\dfrac{x^2+x-6}{x-1}}\)

Simplify \(\dfrac{\dfrac{1}{x}+\dfrac{1}{xy}}{\dfrac{3}{y^2}+\dfrac{1}{y}}\)