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Section 7
Exponents and Order of Operations
In the exponential expression $7^{5}, 7$ is the_____ $\quad,$ and 5 is the____$.7^{5}$ is the fifth_____ $\quad$ of seven.
$10^{2}$ can be read as ten_______ $\quad,$ and $10^{3}$ can be read as ten______.
An__________ is used to represent repeated multiplication.
To____________the expression $2\left(-1+4^{2}\right)$ means to find its value.
The rule for the__________of operations guarantees that an evaluation of a numerical expression will result in a single answer.
To find the arithmetic___________ divide the sum of the values by the number of values.
To evaluate each expression, what operation should be performed first?a. $24-4+2$b. $32 \div 8 \cdot 4$c. $8-(3+5)^{2}$d. $65 \cdot 3^{3}$
To evaluate $\frac{36-4(7)}{2(10-8)},$ what operation should be performed first in the numerator? In the denominator?
a. Give the name of each grouping symbol: $(.),[1,\{]$$|\quad|,$ and $-$b. In the expression $-8+2[15-(-6+1)]$, which grouping symbols are innermost, and which are outermost?
What operation is indicated?$$2+9|5-(2+4)|$$
a. In the expression $(-5)^{2},$ what is the base?b. In the expression $-5^{2},$ what is the base?
Write each expression using symbols. Then evaluate it.a. Negative two squaredb. The opposite of the square of two
Complete the evaluation of each expression.$$\begin{aligned}-19-2\left[(1+2)^{2} \cdot 3\right] &=-19-2\left[^{2} \cdot 3\right] \\&=-19-2[\cdot 3] \\&=-19-2[\quad] \\&=-19-\end{aligned}$$
Complete the evaluation of each expression.$$\begin{aligned}\frac{46-2^{3}}{-3(5)-4} &=\frac{46-}{-4} \\&=\frac{ }{-} \\&=-2\end{aligned}$$
Write each product using exponents. See Example $1 .$$$8 \cdot 8 \cdot 8$$
Write each product using exponents. See Example $1 .$$$(-4)(-4)(-4)(-4)$$
Write each product using exponents. See Example $1 .$$$7 \cdot 7 \cdot 7 \cdot 12 \cdot 12$$
Write each product using exponents. See Example $1 .$$$5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 7 \cdot 7 \cdot 7$$
Write each product using exponents. See Example $1 .$$$x \cdot x \cdot x$$
Write each product using exponents. See Example $1 .$$$b \cdot b \cdot b \cdot b$$
Write each product using exponents. See Example $1 .$$$r \cdot r \cdot r \cdot r \cdot s \cdot s$$
Write each product using exponents. See Example $1 .$$$m \cdot m \cdot m \cdot n \cdot n \cdot n \cdot n$$
Evaluate each expression.$$7^{2}$$
Evaluate each expression.$$9^{2}$$
Evaluate each expression.$$6^{3}$$
Evaluate each expression.$$6^{4}$$
Evaluate each expression.$$(-5)^{4}$$
Evaluate each expression.$$(-5)^{3}$$
Evaluate each expression.$$(-0.1)^{2}$$
Evaluate each expression.$$(-0.8)^{2}$$
Evaluate each expression.$$\left(-\frac{1}{4}\right)^{3}$$
Evaluate each expression.$$\left(-\frac{1}{3}\right)^{4}$$
Evaluate each expression.$$\left(\frac{2}{3}\right)^{3}$$
Evaluate each expression.$$\left(\frac{3}{4}\right)^{3}$$
Evaluate each expression.$$(-6)^{2} \text { and }-6^{2}$$
Evaluate each expression.$$(-4)^{2} \text { and }-4^{2}$$
Evaluate each expression.$$(-8)^{2} \text { and }-8^{2}$$
Evaluate each expression.$$(-9)^{2} \text { and }-9^{2}$$
Evaluate each expression.$$3-5 \cdot 4$$
Evaluate each expression.$$-4 \cdot 6+5$$
Evaluate each expression.$$32-16 \div 4+2$$
Evaluate each expression.$$60-20 \div 10+5$$
Evaluate each expression.$$3^{2} \cdot 5-6 \div 3$$
Evaluate each expression.$$2^{3} \cdot 5-4 \div 2$$
Evaluate each expression.$$12 \div 3 \cdot 2$$
Evaluate each expression.$$18 \div 6 \cdot 3$$
Evaluate each expression.$$-22-15+3$$
Evaluate each expression.$$-33-8+10$$
Evaluate each expression.$$-2(9)-2(5)(10)$$
Evaluate each expression.$$-6(7)-3(-4)(-2)$$
Evaluate each expression.$$-4(6+5)$$
Evaluate each expression.$$-3(5-4)$$
Evaluate each expression.$$(9-3)(9-9)^{2}$$
Evaluate each expression.$$-(-8-6)(6-6)^{2}$$
Evaluate each expression.$$\left(-1-3^{2} \cdot 4\right) 2^{2}$$
Evaluate each expression.$$-1\left(28-5^{2} \cdot 2\right) 3^{2}$$
Evaluate each expression.$$1+5(10+2 \cdot 5)-1$$
Evaluate each expression.$$14+3(7-5 \cdot 3)$$
Evaluate each expression.$$(-1)^{9}\left[-7^{2}-(-2)^{2}\right]$$
Evaluate each expression.$$\left[-9^{2}-(-8)^{2}\right](-1)^{10}$$
Evaluate each expression.$$64-6[15+2(-3+8)]$$
Evaluate each expression.$$4-2[26+2(5-3)]$$
Evaluate each expression.$$-2\left[2+4^{2}(8-9)\right]^{2}$$
Evaluate each expression.$$-3\left[5+3^{2}(4-5)\right]^{2}$$
Evaluate each expression.$$4+2[-7-(3-9)]$$
Evaluate each expression.$$\frac{-2-5}{-7+(-7)}$$
Evaluate each expression.$$\frac{-3-(-1)}{-2+(-2)}$$
Evaluate each expression.$$\frac{2 \cdot 2^{5}-60+(-4)}{5^{4}-(-4)(-5)}$$
Evaluate each expression.$$\frac{(6-5)^{8}-1}{(-9)(-3)-4}$$
Evaluate each expression.$$\frac{2(-4-2 \cdot 2)}{3(-3)(-2)}$$
Evaluate each expression.$$\frac{3\left(-3^{2}+2 \cdot 2^{2}\right)}{(5-8)(7-9)}$$
Evaluate each expression.$$\frac{72-(2-2 \cdot 4)}{10^{2}-\left(9 \cdot 10+2^{2}\right)}$$
Evaluate each expression.$$\frac{13^{2}-5^{2}}{-3\left(5-3^{2}\right)}$$
Evaluate each expression.$$10-2|4-8|$$
Evaluate each expression.$$45-5|1-8|$$
Evaluate each expression.$$-\left|7-2^{3}(4-7)\right|$$
Evaluate each expression.$$-\left|9-5\left(1-2^{3}\right)\right|$$
Evaluate each expression.$$\frac{(3+5)^{2}+|-2|}{-2(5-8)}$$
Evaluate each expression.$$\frac{|-25|-8(-5)}{2^{4}-29}$$
Evaluate each expression.$$\frac{|6-4|+2|-4|}{226-6^{3}}$$
Evaluate each expression.$$\frac{4|9-7|+|-7|}{6^{3}-211}$$
Evaluate each expression.$$-\left(2 \cdot 3-2^{2}\right)^{5}$$
Evaluate each expression.$$-(3 \cdot 5-2 \cdot 6)^{4}$$
Evaluate each expression.$$2 \cdot 5^{2}+4 \cdot 3^{2}$$
Evaluate each expression.$$5 \cdot 3^{3}-4 \cdot 2^{3}$$
Evaluate each expression.$$-2(-1)^{2}+3(-1)-3$$
Evaluate each expression.$$-4(-3)^{2}+3(-3)-1$$
Evaluate each expression.$$8-3\left[5^{2}-(7-3)^{2}\right]$$
Evaluate each expression.$$3-\left[3^{3}+(3-1)^{3}\right]$$
Evaluate each expression.$$[6(5)-5(5)]^{3}(-4)$$
Evaluate each expression.$$5-2 \cdot 3^{4}-(-6+5)^{3}$$
Evaluate each expression.$$8-6\left[\left(130-4^{3}\right)-2\right]$$
Evaluate each expression.$$91-5\left[\left(150-3^{3}\right)-1\right]$$
Evaluate each expression.$$-2\left(\frac{15}{-5}\right)-\frac{6}{2}+9$$
Evaluate each expression.$$-6\left(\frac{25}{-5}\right)-\frac{36}{9}+1$$
Evaluate each expression.$$-5(-2)^{3}-|-2+1|$$
Evaluate each expression.$$6(-3)^{3}-|-6+5|$$
Evaluate each expression.$$\frac{18-[2+(1-6)]}{16-(-4)^{2}}$$
Evaluate each expression.$$\frac{6-[6(-1)-88]}{4-2^{2}}$$
Evaluate each expression.$$-\left|-5 \cdot 7^{2}\right|-30$$
Evaluate each expression.$$2+\left|-3 \cdot 2^{2} \cdot 8^{2} \cdot 1^{2}\right|$$
Evaluate each expression.$$(-3)^{3}\left(\frac{-4}{2}\right)(-1)$$
Evaluate each expression.$$(-2)^{3}\left(\frac{-6}{2}\right)(-1)$$
Evaluate each expression.$$\frac{1}{2}\left(\frac{1}{8}\right)+\left(-\frac{1}{4}\right)^{2}$$
Evaluate each expression.$$-\frac{1}{9}\left(\frac{1}{4}\right)+\left(-\frac{1}{6}\right)^{2}$$
Evaluate each expression.$$\frac{-5^{2} \cdot 10+5 \cdot 2^{5}}{-5-3-1}$$
Evaluate each expression.$$\frac{\left(-6^{2}-2^{4} \cdot 2\right)+5}{-4-3}$$
Evaluate each expression.$$-\left(\frac{40-1^{3}-2^{4}}{3(2+5)+2}\right)$$
Evaluate each expression.$$-\left(\frac{8^{2}-10}{2(3)(4)-5(3)}\right)$$
a. $(-7-4)(-2)$b. $(-7-4)-2$
a. $2 \cdot 3^{3}$b. $(2 \cdot 3)^{3}$
$$\begin{array}{ll}{\text { a. }-100 \div 5 \cdot 2} & {\text { b. }-100 \div(5 \cdot 2)}\end{array}$$
a. $8+3[-2-(6+1)]$b. $(8+3)[-2-(6+1)]$
Light. As light energy passes through the first unit of area, 1 yard away from the bulb, it spreads out. How much area does that light energy cover 2 yards, 3 yards, and 4 yards from the bulb? Express each answer using exponents.
Chain Letters. A woman sent two friends a letter with the following request: "Please send a copy of this letter to two of your friends." Assume that all those receiving letters responded and that everyone in the chain received just one letter. Complete the table and then determine how many letters will be circulated in the 10 th level.$$\begin{array}{|c|c|}\hline \text { Level } & {\text { Number of letters circulated }} \\\hline \text { 1st } & {2=2^{1}} \\\hline 2 \mathrm{nd} & {=2} \\\hline 3 \mathrm{rd} & {=2} \\\hline 4 \mathrm{th} & {=2} \\\hline\end{array}$$
To determine the average afternoon wait time in security lines at an airport, officials monitored four passengers, each at a different gate. The time that each passenger entered a security line and the time the same passenger cleared the checkpoint was recorded, as shown below. Find the average (mean) wait time for these passengers.$$\begin{array}{|l|c|c|}\hline & {\text { Time }} & {\text { Time }} \\\hline & {\text { entered }} & {\text { cleared }} \\\hline \text { Passenger at Gate A } & {3: 05 \text { pm }} & {3: 21 \text { pm }} \\\hline \text { Passenger at Gate B } & {3: 03 \text { pm }} & {3: 13 \text { pm }} \\\hline \text { Passenger at Gate C } & {3: 01 \text { pm }} & {3: 09 \text { pm }} \\\hline \text { Passenger at Gate D } & {3: 02 \text { pm }} & {3: 16 \text { pm }} \\\hline\end{array}$$
Energy Usage. Find the average number of therms of natural gas used per month.
Cash Awards. A contest is to be part of a promotional kickoff for a new children's cereal. The prizes to be awarded are shown.a. How much money will be awarded in the promotion?b. What is the average cash prize?
Surveys. Some students were asked to rate their college cafeteria food on a scale from 1 to 5. The responses are shown on the tally sheet. Find the average rating.
Wrapping Gifts. How much ribbon is needed to wrap the package if 15 inches of ribbon are needed to make the bow?
Scrabble. Write an expression to determine the number of points received for playing the word QUARTZY and then evaluate it. (The number on each tile gives the point value of the letter.
Explain the difference between $2^{3}$ and $3^{2}$.
Why is the order of operations rule necessary?
Explain the error. What is the correct answer?
What numbers are a distance of 6 away from $-11$ on a number line?
Fill in the blank with $>$ or $<: 0.3 \quad \frac{1}{3}$
Using each of the numbers $2,3,$ and 4 only once, what is the greatest value that the following expression can have?
Insert a pair of parentheses into $4 \cdot 3^{2}-4 \cdot 2$ so that it has a value of 40
Translate the set of instructions to an expression and then evaluate it.Subtract the sum of $-9$ and 8 from the product of the cube of $-3$ and the opposite of 4
Translate the set of instructions to an expression and then evaluate it.Increase the square of the reciprocal of $-2$ by the difference of $-0.25$ and $-1$
