Chapter 1

According to the U.S. Mint, the diameter of a quarter is 0.955 inches. The circumference of the quarter would be the diameter multiplied by \(\pi\). Is the circumference of a quarter a whole number, a rational number, or an irrational number?

For the following exercises, perform the given operations and simplify. \(\frac{\frac{4 a+1}{2 a-3}+\frac{2 a-3}{2 a+3}}{\frac{4 a^{2}+9}{a}}\)

For the following exercises, factor the polynomials completely. \(81 y^{4}-256\)

For the following exercises, simplify each expression. \(\sqrt{\frac{42 q}{36 q^{3}}}\)

For the following exercises, simplify the given expression. Write answers with positive exponents. \(\left(\frac{x^{6} y^{3}}{x^{3} y^{-3}} \cdot \frac{y^{-7}}{x^{-3}}\right)^{10}\)

Jessica and her roommate, Adriana, have decided to share a change jar for joint expenses. Jessica put her loose change in the jar first, and then Adriana put her change in the jar. We know that it does not matter in which order the change was added to the jar. What property of addition describes this fact?

For the following exercises, perform the given operations and simplify. \(\frac{x^{2}+7 x+12}{x^{2}+x-6} \div \frac{3 x^{2}+19 x+28}{8 x^{2}-4 x-24} \div \frac{2 x^{2}+x-3}{3 x^{2}+4 x-7}\)

For the following exercises, factor the polynomials completely. \(16 z^{4}-2,401 a^{4}\)

For the following exercises, simplify each expression. \(\sqrt{\frac{81 m}{361 m^{2}}}\)

For the following exercises, simplify the given expression. Write answers with positive exponents. \(\left(\frac{\left(a b^{2} c\right)^{-3}}{b^{-3}}\right)^{2}\)

For the following exercises, simplify each expression. \(\sqrt{72 c}-2 \sqrt{2 c}\)

Avogadro's constant is used to calculate the number of particles in a mole. A mole is a basic unit in chemistry to measure the amount of a substance. The constant is \(6.0221413 \times 10^{23} .\) Write Avogadro's constant in standard notation.

For the following exercises, consider this scenario: There is a mound of g pounds of gravel in a quarry. Throughout the day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel. Solve for g.

For the following exercises, factor the polynomials completely. \(\left(32 x^{3}+48 x^{2}-162 x-243\right)^{-1}\)

For the following exercises, simplify each expression. \(\sqrt{\frac{144}{324 d^{2}}}\)

Planck's constant is an important unit of measure in quantum physics. It describes the relationship between energy and frequency. The constant is written as \(6.62606957 \times 10^{-34} .\) Write Planck's constant in standard notation.

Ramon runs the marketing department at his company. His department gets a budget every year, and every year, he must spend the entire budget without going over. If he spends less than the budget, then his department gets a smaller budget the following year. At the beginning of this year, Ramon got \(\$ 2.5\) million for the annual marketing budget. He must spend the budget such that \(2,500,000-x=0 .\) What property of addition tells us what the value of \(x\) must be?

For the following exercises, simplify each expression. \(\sqrt[3]{24 x^{6}}+\sqrt[3]{81 x^{6}}\)

For the following exercises, use a graphing calculator to solve for \(x\). Round the answers to the nearest hundredth. \(0.5(12.3)^{2}-48 x=\frac{3}{5}\)

For the following exercises, simplify each expression. \(\sqrt[4]{\frac{162 x^{6}}{16 x^{4}}}\)

For the following exercises, use a graphing calculator to solve for \(x\). Round the answers to the nearest hundredth. \((0.25-0.75)^{2} x-7.2=9.9\)

For the following exercises, simplify each expression. \(\sqrt[3]{64 y}\)

If a whole number is not a natural number, what must the number be?

For the following exercises, simplify each expression. \(\sqrt[3]{128 z^{3}}-\sqrt[3]{-16 z^{3}}\)

Determine whether the statement is true or false: The multiplicative inverse of a rational number is also rational.

For the following exercises, simplify each expression. \(\sqrt[5]{1,024 c^{10}}\)

Determine whether the statement is true or false: The product of a rational and irrational number is always irrational.

A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So the length of the guy wire can be found by evaluating \(\sqrt{90,000+160,000}\). What is the length of the guy wire?

Determine whether the simplified expression is rational or irrational: \(\sqrt{-18-4(5)(-1)}\)

A car accelerates at a rate of \(6-\frac{\sqrt{4}}{\sqrt{t}} \mathrm{~m} / \mathrm{s}^{2}\) where \(t\) is the time in seconds after the car moves from rest. Simplify the expression.

Determine whether the simplified expression is rational or irrational: \(\sqrt{-16+4(5)+5} .\)

For the following exercises, simplify each expression. \(\frac{\sqrt{8}-\sqrt{16}}{4-\sqrt{2}}-2^{\frac{1}{2}}\)

For the following exercises, simplify each expression. \(\frac{4^{\frac{3}{2}}-16^{\frac{3}{2}}}{8^{\frac{1}{3}}}\)

What property of real numbers would simplify the following expression: \(4+7(x-1) ?\)

For the following exercises, simplify each expression. \(\frac{\sqrt{m n^{3}}}{a^{2} \sqrt{c^{-3}}} \cdot \frac{a^{-7} n^{-2}}{\sqrt{m^{2} c^{4}}}\)

For the following exercises, simplify each expression. \(\frac{a}{a-\sqrt{c}}\)

For the following exercises, simplify each expression. \(\frac{x \sqrt{64 y}+4 \sqrt{y}}{\sqrt{128 y}}\)

For the following exercises, simplify each expression. \(\left(\frac{\sqrt{250 x^{2}}}{\sqrt{100 b^{3}}}\right)\left(\frac{7 \sqrt{b}}{\sqrt{125 x}}\right)\)

For the following exercises, simplify each expression. \(\sqrt{\frac{\sqrt[3]{64}+\sqrt[4]{256}}{\sqrt{64}+\sqrt{256}}}\)