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

Solve the following equation for \(x\).
\(3(x+2)-2=4x\)

Solve the equation:

\(\dfrac{x+5}{2}+\dfrac{2x-1}{5}=5\)

Solve the equation:

\(0.3(x+2)-0.02x=0.5\)

The perimeter of a rectangular fence is \(15\) feet. Write the width of the fence in terms of the length.

Solve \(2x^2-7x+3=0\) by factoring.

Solve \(-3x^2+9=0\)

Solve the equation \(3x^2-6x-1=0\) by using the technique of completing the square.

Solve the equation \(-4x^2+3x+\dfrac{1}{2}=0\) by using the quadratic formula.

Use the quadratic formula to find the real solutions of the equation \(-2t^2+3t=5\). Find the value of the discriminant.

The height of a ball after being thrown vertically upward from a point \(80\) feet above the ground with a velocity of \(40\) feet per second is given by \(h=-16t^2+40t+80\), where \(t\) is the time in seconds since the ball was thrown and \(h\) is in feet.

1. When will the ball be \(50\) feet above the ground?

2. When will the ball reach the ground?

3. For what values of \(t\) does this problem make sense (from a physical sense)?

Solve \(x-4>-2x+2\)

Alicia has a total of \(\$1\text{,}000\) to spend on a new computer system. If the sales tax is \(8\%\), what is the retail price range of computers that she should consider?

To operate a gourmet coffee booth in a shopping mall, it costs \(\$500\) (the fixed cost) plus \(\$6\) for each pound of coffee bought at wholesale price. The coffee is then sold to customers for \(\$10\) per pound.

1. Find an expression for the operating cost of selling \(q\) pounds of coffee.

2. Find an expression for the revenue earned by selling \(q\) pounds of coffee.

3. Find the break-even point.

4. How many pounds of coffee must be sold for the revenue to be greater than the total cost?

Solve the equation \(3x^4+5x^2-2=0\) for real values of \(x\).

Solve \(\sqrt{3x+1}+2=x-1\)

Solve \(\sqrt{3x+1}-\sqrt{x+4}=1\)