0. Review of College Algebra

CONCEPT PREVIEW Perform the operations mentally, and write the answers without doing intermediate steps. √25 + √64

CONCEPT PREVIEW Perform the operations mentally, and write the answers without doing intermediate steps. √6 • √6

CONCEPT PREVIEW Perform the operations mentally, and write the answers without doing intermediate steps. (√28 - √14) (√28 + √14)

Find each square root. See Example 1. √100

Find each square root. See Example 1. -√256

Find each square root. See Example 1. √4⁄25

Find each square root. See Example 1. -√144⁄121

Find each square root. See Example 1. √-121

Find the square of each radical expression. See Example 2. -√19

Find the square of each radical expression. See Example 2. √2 3

Find the square of each radical expression. See Example 2. √3x² + 4

​​​​​​​​​​Find each root. See Example 3.​​ -∛512

​​​​​​​​​​Find each root. See Example 3.​​ -∜16

​​​​​​​​​​Find each root. See Example 3.​​ ∛0.001

​​​​​​​​​​Find each root. See Example 3.​​ ∛0.125

Use the product and quotient rules for radicals to rewrite each expression. See Example 4. √3 • √5

Use the product and quotient rules for radicals to rewrite each expression. See Example 4. √3 • √27

Use the product and quotient rules for radicals to rewrite each expression. See Example 4. √7⁄16

Use the product and quotient rules for radicals to rewrite each expression. See Example 4. √4⁄50

Use the product and quotient rules for radicals to rewrite each expression. See Example 4. √5 √20

Use the product and quotient rules for radicals to rewrite each expression. See Example 4. 30√10 5√2

Simplify each radical. See Example 5. √24

Simplify each radical. See Example 5. √75

Simplify each radical. See Example 5. - √160

Simplify each radical. See Example 5. 3√27

Add or subtract, as indicated. See Example 6. 2√3 + 5√3

Add or subtract, as indicated. See Example 6. 5√3 - √3

Add or subtract, as indicated. See Example 6. √6 + √6

Add or subtract, as indicated. See Example 6. √6 + √7

Add or subtract, as indicated. See Example 6. 5√3 + √12

Add or subtract, as indicated. See Example 6. √45 + 4√20

Add or subtract, as indicated. See Example 6. 2√50 - 5√72

Add or subtract, as indicated. See Example 6. -5√32 + 2√98

Multiply. See Example 7. √6 (3 + √2)

Multiply. See Example 7. (√2 + 1) (√3 + 1)

Multiply. See Example 7. (√2 - √3) (√2 + √3)

Multiply. See Example 7. (√5 + 2)²

Rationalize each denominator. See Example 8. 6 —— √5

Rationalize each denominator. See Example 8. 5 —— √5

Rationalize each denominator. See Example 8. 4 —— √6

Rationalize each denominator. See Example 8. 18 —— √27

Rationalize each denominator. See Example 8. 12 —— √72

Rationalize each denominator. See Example 8. 3 ———— 4 + √5

Rationalize each denominator. See Example 8. 6 ———— √5 + √3

Rationalize each denominator. See Example 8. √3 + 1 ———— 1 - √3

Rationalize each denominator. See Example 8. √2 - √3 ———— √6 - √5

Simplify. See Example 9. - √2 3 ——— √7 3

Simplify. See Example 9. √7 5 ——— √3 10

Simplify. See Example 9. 1 2 ——— 1 - √5 2

Simplify. See Example 9. √3 2 ——— 1 - √3 2

For Individual or Group Work (Exercises 147 – 150)In calculus, it is sometimes desirable to rationalize a numerator. To do this, we multiply the numerator and the denominator by the conjugate of the numerator. For example, 6 - √2 = 6 - √2 • 6 + √2 = 36 - 2 = 34 = 17 . 4 4 6 + √2 4(6 + √2) 4(6 + √2) 2(6 + √2) Rationalize each numerator. 6 - √3 8

For Individual or Group Work (Exercises 147 – 150)In calculus, it is sometimes desirable to rationalize a numerator. To do this, we multiply the numerator and the denominator by the conjugate of the numerator. For example, 6 - √2 = 6 - √2 • 6 + √2 = 36 - 2 = 34 = 17 . 4 4 6 + √2 4(6 + √2) 4(6 + √2) 2(6 + √2) Rationalize each numerator. 2√10 + √7 30

CONCEPT PREVIEW Perform the indicated operation, and write each answer in lowest terms 2x 10x —— • ——— 5 x²

CONCEPT PREVIEW Perform the indicated operation, and write each answer in lowest terms 3 7 —— + —— x x

CONCEPT PREVIEW Perform the indicated operation, and write each answer in lowest terms 2x x —— + —— 5 4

Find the domain of each rational expression. See Example 1. x + 3 x - 6

Find the domain of each rational expression. See Example 1. 3x + 7 ——————— (4x + 2) (x - 1)

Find the domain of each rational expression. See Example 1. 12 —————— x² + 5x + 6

Find the domain of each rational expression. See Example 1. x² - 1 ———— x + 1

Find the domain of each rational expression. See Example 1. x³ - 1 ———— x - 1

Write each rational expression in lowest terms. See Example 2. 8x² + 16x 4x²

Write each rational expression in lowest terms. See Example 2. 3 (3 - t) —————— (t + 5) (t - 3)

Write each rational expression in lowest terms. See Example 2. 8k + 16 9k + 18

Write each rational expression in lowest terms. See Example 2. m² - 4m + 4 m² + m - 6

Write each rational expression in lowest terms. See Example 2. 8m² + 6m - 9 16m² - 9

Multiply or divide, as indicated. See Example 3. 15p³ 12p —— • ——— 9p² 10p³

Multiply or divide, as indicated. See Example 3. 2k + 8 3k + 12 ———— ÷ ———— 6 2

Multiply or divide, as indicated. See Example 3. x² + x 25 ———— • ———— 5 xy + y

Multiply or divide, as indicated. See Example 3. 4a + 12 a² - 9 ————— ÷ ————— 2a - 10 a² - a - 20

Multiply or divide, as indicated. See Example 3. m² + 3m + 2 m² + 5m + 6 ——————— ÷ ———————— m² + 5m + 4 m² + 10m + 24

Multiply or divide, as indicated. See Example 3. xz - xw + 2yz - 2yw 4z + 4w +xz +wx —————————— • ————————— z² - w² 16 - x²

Add or subtract, as indicated. See Example 4. 3 5 —— + —— 2k 3k

Add or subtract, as indicated. See Example 4. 1 2 4 —— + —— + —— 6m 5m m

Add or subtract, as indicated. See Example 4. 1 b —— + —— a a²

Add or subtract, as indicated. See Example 4. 5 11 ———— - ——— 12x²y 6xy

Add or subtract, as indicated. See Example 4. 17y + 3 -10y - 18 ———— - ————— 9y + 7 9y + 7

Add or subtract, as indicated. See Example 4. 1 1 ——— + ——— x + z x - z

Add or subtract, as indicated. See Example 4. 3 1 ——— - ——— a - 2 2 - a

Add or subtract, as indicated. See Example 4. x + y 2x ——— - ——— 2x - y y - 2x

Add or subtract, as indicated. See Example 4. 4 1 12 ———— + —————— - ———— x + 1 x² - x + 1 x³ + 1

Add or subtract, as indicated. See Example 4. 3x x —————— - ———— x² + x - 12 x² - 16

Simplify each complex fraction. See Examples 5 and 6. 4 - — 3 ———— 2 — 9

Simplify each complex fraction. See Examples 5 and 6. y r —— x r

Simplify each complex fraction. See Examples 5 and 6. 5⁄8 + 2⁄3 ———— 7⁄3 - 1⁄4

Simplify each complex fraction. See Examples 5 and 6. ( -4⁄3 ) + 12⁄5 ———————— 1 - ( -4⁄3 ) (12⁄5)

Simplify each complex fraction. See Examples 5 and 6. 1 + 1 x ———— 1 - 1 x

Simplify each complex fraction. See Examples 5 and 6. x y — + — y x —————— x y — - — y x

Simplify each complex fraction. See Examples 5 and 6. 1 1 ——— - —— x + 1 x ———————— 1 —— x

Simplify each complex fraction. See Examples 5 and 6. y + 3 4 ———— - ———— y y - 1 —————————— y 1 ——— + —— y - 1 y

(Modeling) Distance from the Origin of the Nile River The Nile River in Africa is about 4000 mi long. The Nile begins as an outlet of Lake Victoria at an altitude of 7000 ft above sea level and empties into the Mediterranean Sea at sea level (0 ft). The distance from its origin in thousands of miles is related to its height above sea level in thousands of feet (x) by the following formula. 7 - x Distance = ——————— 0.639x + 1.75 For example, when the river is at an altitude of 600 ft, x = 0.6 (thousand), and the distance from the origin is 7 - 0.6 Distance = ————————— ≈ 3, which represents 3000 mi. 0.639(0.6) + 1.75 (Data from The World Almanac and Book of Facts.) What is the distance from the origin of the Nile when the river has an altitude of 7000 ft?