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. $4\cdot 1=4$

Evaluate the following.

1. $\lvert -3\rvert$

2. $-4-\lvert 4.5\rvert$

3. $\lvert -4+9\rvert +3$

Simplify each expression.

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

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

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

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

1. $4+10\div 5+2$

2. $\dfrac{4+10}{5+2}$

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 $(-2-3)(5-9)$

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

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

Simplify: $\dfrac{-8-8}{-5-3}$

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 $\dfrac{\dfrac{1}{x}+\dfrac{1}{xy}}{\dfrac{3}{y^2}+\dfrac{1}{y}}$