1.8.1: Review Exercises

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Review Exercises

Use the Language of Algebra

Identify Multiples and Factors

385.

Use the divisibility tests to determine whether 180 is divisible by 2, by 3, by 5, by 6, and by 10.

386.

Find the prime factorization of 252.

387.

Find the least common multiple of 24 and 40.

In the following exercises, simplify each expression.

388.

$24 \div 3 + 4 (5 - 2)$

389.

$7 + 3 [6 - 4 (5 - 4)] - 3^{2}$

Evaluate an Expression

In the following exercises, evaluate the following expressions.

390.

When $x = 4,$ ⓐ $x^{3}$ ⓑ $5^{x}$ ⓒ $2 x^{2} - 5 x + 3$

391.

$2 x^{2} - 4 x y - 3 y^{2}$ when $x = 3,$ $y = 1$

Simplify Expressions by Combining Like Terms

In the following exercises, simplify the following expressions by combining like terms.

392.

$12 y + 7 + 2 y - 5$

393.

$14 x^{2} - 9 x + 11 - 8 x^{2} + 8 x - 6$

Translate an English Phrase to an Algebraic Expression

In the following exercises, translate the phrases into algebraic expressions.

394.

ⓐ the sum of $4 a b^{2}$ and $7 a^{3} b^{2}$
ⓑ the product of $6 y^{2}$ and $3 y$
ⓒ twelve more than $5 x$
ⓓ $5 y$ less than $8 y^{2}$

395.

ⓐ eleven times the difference of $y$ and two
ⓑ the difference of eleven times $y$ and two

396.

Dushko has nickels and pennies in his pocket. The number of pennies is four less than five times the number of nickels. Let $n$ represent the number of nickels. Write an expression for the number of pennies.

Integers

Simplify Expressions with Absolute Value

In the following exercise, fill in $<, >,$ or $=$ for each of the following pairs of numbers.

397.

ⓐ $− | 7 |___- | −7 |$
ⓑ $−8___- | −8 |$
ⓒ $| −13 |___- 13$
ⓓ $| −12 |___- (−12)$

In the following exercises, simplify.

398.

$9 - | 3 (4 - 8) |$

399.

$12 - 3 | 1 - 4 (4 - 2) |$

Add and Subtract Integers

In the following exercises, simplify each expression.

400.

$−12 + (−8) + 7$

401.

ⓐ $15 - 7$
ⓑ $−15 - (−7)$
ⓒ $−15 - 7$
ⓓ $15 - (−7)$

402.

$−11 - (−12) + 5$

403.

ⓐ $23 - (−17)$ ⓑ $23 + 17$

404.

$− (7 - 11) - (3 - 5)$

Multiply and Divide Integers

In the following exercise, multiply or divide.

405.

ⓐ $−27 \div 9$ ⓑ $120 \div (−8)$ ⓒ $4 (−14)$ ⓓ $−1 (−17)$

Simplify and Evaluate Expressions with Integers

In the following exercises, simplify each expression.

406.

ⓐ ${(−7)}^{3}$ ⓑ $− 7^{3}$

407.

$(7 - 11) (6 - 13)$

408.

$63 \div (−9) + (−36) \div (−4)$

409.

$6 - 3 | 4 (1 - 2) - (7 - 5) |$

410.

${(−2)}^{4} - 24 \div (13 - 5)$

For the following exercises, evaluate each expression.

411.

${(y + z)}^{2}$ when
$y = −4, z = 7$

412.

$3 x^{2} - 2 x y + 4 y^{2}$ when
$x = −2, y = −3$

Translate English Phrases to Algebraic Expressions

In the following exercises, translate to an algebraic expression and simplify if possible.

413.

the sum of $−4$ and $−9,$ increased by 23

414.

ⓐ the difference of 17 and $−8$ ⓑ subtract 17 from $−25$

Use Integers in Applications

In the following exercise, solve.

415.

Temperature On July 10, the high temperature in Phoenix, Arizona, was 109°, and the high temperature in Juneau, Alaska, was 63°. What was the difference between the temperature in Phoenix and the temperature in Juneau?

Fractions

Simplify Fractions

In the following exercises, simplify.

416.

$\frac{204}{228}$

417.

$- \frac{270 x^{3}}{198 y^{2}}$

Multiply and Divide Fractions

In the following exercises, perform the indicated operation.

418.

$(- \frac{14}{15}) (\frac{10}{21})$

419.

$\frac{6 x}{25} \div \frac{9 y}{20}$

420.

$\frac{- \frac{4}{9}}{\frac{8}{21}}$

Add and Subtract Fractions

In the following exercises, perform the indicated operation.

421.

$\frac{5}{18} + \frac{7}{12}$

422.

$\frac{11}{36} - \frac{15}{48}$

423.

ⓐ $\frac{5}{8} + \frac{3}{4}$ ⓑ $\frac{5}{8} \div \frac{3}{4}$

424.

ⓐ $- \frac{3 y}{10} - \frac{5}{6}$ ⓑ $- \frac{3 y}{10} \cdot \frac{5}{6}$

Use the Order of Operations to Simplify Fractions

In the following exercises, simplify.

425.

$\frac{4 \cdot 3 - 2 \cdot 5}{−6 \cdot 3 + 2 \cdot 3}$

426.

$\frac{4 (7 - 3) - 2 (4 - 9)}{−3 (4 + 2) + 7 (3 - 6)}$

427.

$\frac{4^{3} - 4^{2}}{{(\frac{4}{5})}^{2}}$

Evaluate Variable Expressions with Fractions

In the following exercises, evaluate.

428.

$4 x^{2} y^{2}$ when
$x = \frac{2}{3}$ and $y = - \frac{3}{4}$

429.

$\frac{a + b}{a - b}$ when
$a = −4,$ $b = 6$

Decimals

Round Decimals

430.

Round $6.738$ to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.

Add and Subtract Decimals

In the following exercises, perform the indicated operation.

431.

$−23.67 + 29.84$

432.

$54.3 - 100$

433.

$79.38 - (−17.598)$

Multiply and Divide Decimals

In the following exercises, perform the indicated operation.

434.

$(−2.8) (3.97)$

435.

$(−8.43) (−57.91)$

436.

$(53.48) (10)$

437.

$(0.563) (100)$

438.

$$ 118.35 \div 2.6$

439.

$1.84 \div (−0.8)$

Convert Decimals, Fractions and Percents

In the following exercises, convert each decimal to a fraction.

440.

$0.65$

441.

$−9.6$

In the following exercises, convert each fraction to a decimal.

442.

$- \frac{5}{8}$

443.

$\frac{14}{11}$

In the following exercises, convert each decimal to a percent.

444.

$2.43$

445.

$0.0475$

Simplify Expressions with Square Roots

In the following exercises, simplify.

446.

$\sqrt{289}$

447.

$\sqrt{−121}$

Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

In the following exercise, list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers for each set of numbers

448.

$−8, 0, 1.95286..., \frac{12}{5}, \sqrt{36}, 9$

Locate Fractions and Decimals on the Number Line

In the following exercises, locate the numbers on a number line.

449.

$\frac{3}{4}, - \frac{3}{4}, 1 \frac{1}{3}, −1 \frac{2}{3}, \frac{7}{2}, - \frac{5}{2}$

450.

ⓐ $3.2$ ⓑ $−1.35$

Properties of Real Numbers

Use the Commutative and Associative Properties

In the following exercises, simplify.

451.

$\frac{5}{8} x + \frac{5}{12} y + \frac{1}{8} x + \frac{7}{12} y$

452.

$−32 \cdot 9 \cdot \frac{5}{8}$

453.

$(\frac{11}{15} + \frac{3}{8}) + \frac{5}{8}$

Use the Properties of Identity, Inverse and Zero

In the following exercises, simplify.

454.

$\frac{4}{7} + \frac{8}{15} + (- \frac{4}{7})$

455.

$\frac{13}{15} \cdot \frac{9}{17} \cdot \frac{15}{13}$

456.

$\frac{0}{x - 3}, x \neq 3$

457.

$\frac{5 x - 7}{0}, 5 x - 7 \neq 0$

Simplify Expressions Using the Distributive Property

In the following exercises, simplify using the Distributive Property.

458.

$8 (a - 4)$

459.

$12 (\frac{2}{3} b + \frac{5}{6})$

460.

$18 \cdot \frac{5}{6} (2 x - 5)$

461.

$(x - 5) p$

462.

$−4 (y - 3)$

463.

$12 - 6 (x + 3)$

464.

$6 (3 x - 4) - (−5)$

465.

$5 (2 y + 3) - (4 y - 1)$

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Use the Language of Algebra