1.1.1: Exercises 1.1

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Terms and Concepts

Exercise $\PageIndex{1}$

In your own words, what does “multiplication and division are on the same level” mean?

Answer: Answers will vary.

Exercise $\PageIndex{2}$

In an expression with both addition and multiplication, which operation do you complete first?

Answer: Multiplication

Exercise $\PageIndex{3}$

In your own words, what is meant by “implied parentheses”? Provide an example.

Answer: Answers will vary.

Exercise $\PageIndex{4}$

T/F: In an expression with only addition and subtraction remaining, you must complete all of the addition before starting the subtraction. Explain.

Answer: F; you should complete them from left to right

Exercise $\PageIndex{5}$

T/F: In the expression $−2^{2}$, only “2” is squared, not “−2.” Explain.

Answer: T; there are no parentheses, so you first square $2$ and then make the answer negative.

Problems

In exercises $\PageIndex{6}$-$\PageIndex{15}$, simplify the given expressions.

Exercise $\PageIndex{6}$

$\displaystyle -2(11-5) \div 3 + 2^3$

Answer: $4$

Exercise $\PageIndex{7}$

$\frac{3}{5}+\frac{2}{3}\div\frac{5}{6}$

Answer: $\frac{7}{5}$

Exercise $\PageIndex{8}$

$\left(\frac{2}{3}\right)^{2}-\frac{1}{6}$

Answer: $\frac{5}{18}$

Exercise $\PageIndex{9}$

$\displaystyle (13+7) \div 4 -3^2$

Answer: $-4$

Exercise $\PageIndex{10}$

$\left(5-\frac{1}{2}\right)^{3}$

Answer: $\frac{729}{8}$

Exercise $\PageIndex{11}$

$(2)(-2)\div\frac{1}{2}$

Answer: $-8$

Exercise $\PageIndex{12}$

$\displaystyle \frac{2-4(3-5)}{6-7+3} -\sqrt{25-9}$

Answer: $1$

Exercise $\PageIndex{13}$

$\displaystyle \sqrt{\frac{2^2+3^2+5}{2-(-1)^3} -2}+6$

Answer: $8$

Exercise $\PageIndex{14}$

$\displaystyle \frac{4-1(-2)}{\frac{1}{2}+1}-2$

Answer: $2$

Exercise $\PageIndex{15}$

$-4^{2}+5^{2}-2\times 3+1$

Answer: $4$

In exercises $\PageIndex{16}$ – $\PageIndex{21}$, evaluate the described mathematical statement, or determine how the described changes affect other variables in the statement as appropriate.

Exercise $\PageIndex{16}$

A runner leaves her home and runs straight for 30 minutes at a pace of 1 mi every 10 minutes (a 10-minute mile). She makes a 90-degree left turn and then runs straight for 40 minutes at the same pace. What is the distance between her current location and her home?

Answer: 5 miles

Exercise $\PageIndex{17}$

The Reynold’s number, which helps identify whether or not a fluid flow is turbulent, is given by $Re=\frac{\rho uD}{\mu}$. If $\rho$, $u$, and $D$ are held constant while $\mu$ increases, does $Re$ increase, decrease, or stay the same?

Answer: $Re$ decreases.

Exercise $\PageIndex{18}$

Consider a square-based box whose height is twice the length of the base of the side. If the length of the square base is 3 ft, what is the volume of the box? (Don’t forget your units!)

Answer: 54 ft$^3$

Exercise $\PageIndex{19}$

The velocity of periodic waves, $v$, is given by $v=\lambda f$ where $\lambda$ is the length of the waves and $f$ is the frequency of the waves. If the wavelength is held constant while the frequency is tripled, what happens to the velocity of the waves? Be as descriptive as possible.

Answer: The velocity is tripled.

Exercise $\PageIndex{20}$

The capacitance, $C$, of a parallel plate capacitor is given by $C=\frac{k \epsilon_0 A}{d}$ where $d$ is the distance between the plates. If $k$, $\epsilon_0$, and $A$ are held constant while the distance between the plates decreases, does the capacitance increase, decrease, or stay the same?

Answer: The capacitance increases.

Exercise $\PageIndex{21}$

Consider a square-based box whose height is half the length of the sides for the base. If the surface area of the base is 16 ft$^2$, what is the volume of the box?

Answer: 32 ft$^3$

In exercises $\PageIndex{22}$ – $\PageIndex{25}$, evaluate the given work for correctness. Is the final answer correct? If not, describe the error(s) in the solution.

Exercise $\PageIndex{22}$

\[\begin{split} 12 \div 6 \times 4 - (3-4)^2 &= 12 \div 6 \times 4 - (-1)^2 \\ &= 12 \div 6 \times 4 +1 \\ &= 12 \div 24 + 1 \\ &= \frac{1}{2} + 1 \\ &= \frac{3}{2} \end{split}\]

Answer: The final answer is not correct. There are two errors: in the second step, $-(-1)^2$ should give $-1$, not $+1$; in the third step, the division should have been done before the multiplication.

Exercise $\PageIndex{23}$

\[\begin{split} -3^2 + 6 \div 2 + (-4)^2 & = -9 + 6 \div 2 + (-4)^2 \\ & = -9 + 6 \div 2 + 16 \\ & = -9 + 3 + 16 \\ & = 10 \end{split}\]

Answer: The final answer is correct.

Exercise $\PageIndex{24}$

\[\begin{split} \frac{2+(3-5)}{2-2(6+3)} + \sqrt{64+36} & = \frac{2+(-2)}{2-2(6+3)} + \sqrt{64+36} \\ & = \frac{2-2}{2-2(9)} + \sqrt{64+36} \\ & = \frac{2-2}{2-2(9)} + 8+6 \\ & = \frac{2-2}{2-18} +8 + 6 \\ & = \frac{0}{-16} + 14 \\ & = 14 \end{split}\]

Answer: The final answer is not correct. In the square root, 64 and 36 should be added before the square root is taken.

Exercise $\PageIndex{25}$

\[\begin{split} \frac{(-3+1)(2(-3))-((-3)^2-3)(1)}{(-3+1)^2} & = \frac{(-3+1)(2(-3)) - ((-3)^2-3)(1)}{(-2)^2} \\ & = \frac{(-2)(-6)-((-3)^2-3)(1)}{(-2)^2} \\ & = \frac{(-2)(-6) -(9-3)(1)}{(-2)^2} \\ & = \frac{(-2)(-6) -(6)(1)}{4} \\ & = \frac{-12-6}{4} \\ & = \frac{-18}{4} \\ & = \frac{-9}{2} \end{split}\]

Answer: The final answer is not correct. In the fifth step, $(-2)(-6)$ should be $12$, not $-12$.