Chapter 7

Use rational expressions to write as a single radical expression. $$ \sqrt{5 r} \cdot \sqrt[3]{s} $$

Solve. \(3 \sqrt{x^{2}-8 x}=x^{2}-8 x\)

Find the midpoint of the line segment whose endpoints are given. See Example 7 $$ (4.6,-3.5),(7.8,-9.8) $$

Simplify each exponential expression. $$ \frac{\left(2 a^{-1} b^{2}\right)^{3}}{\left(8 a^{2} b\right)^{-2}} $$

Use rational expressions to write as a single radical expression. $$ \sqrt[3]{b} \cdot \sqrt[5]{4 a} $$

Solve. \(\sqrt{\left(x^{2}-x\right)+7}=2\left(x^{2}-x\right)-1\)

$$ \left(5 x^{4}-3 x^{2}+2\right) \div(x+2) $$

Find the midpoint of the line segment whose endpoints are given. See Example 7 $$ (-4.6,2.1),(-6.7,1.9) $$

Determine whether the following are real numbers. $$ \sqrt{-17} $$

2 because 9 is a perfect square. $$ 75 $$ # Write each integer as a product of two integers such that one of the factors is a perfect square. For example, write 18 as 9 # 2 because 9 is a perfect square. $$ 75 $$

Solve. \(7-\left(x^{2}-3 x\right)=\sqrt{\left(x^{2}-3 x\right)+5}\)

Perform each indicated operation. See Sections 1.4 and 5.4 $$ 6 x+8 x $$

Determine whether the following are real numbers. $$ \sqrt[3]{-17} $$

2 because 9 is a perfect square. $$ 20 $$ # Write each integer as a product of two integers such that one of the factors is a perfect square. For example, write 18 as 9 # 2 because 9 is a perfect square. $$ 20 $$

Solve. \(x^{2}+6 x=4 \sqrt{x^{2}+6 x}\)

Perform each indicated operation. See Sections 1.4 and 5.4 $$ (6 x)(8 x) $$

Determine whether the following are real numbers. $$ \sqrt[10]{-17} $$

2 because 9 is a perfect square. $$ 48 $$ # Write each integer as a product of two integers such that one of the factors is a perfect square. For example, write 18 as 9 # 2 because 9 is a perfect square. $$ 48 $$

Perform each indicated operation. See Sections 1.4 and 5.4 $$ (2 x+3)(x-5) $$

Determine whether the following are real numbers. $$ \sqrt[15]{-17} $$

2 because 9 is a perfect square. $$ 45 $$ # Write each integer as a product of two integers such that one of the factors is a perfect square. For example, write 18 as 9 # 2 because 9 is a perfect square. $$ 45 $$

Perform each indicated operation. See Sections 1.4 and 5.4 $$ (2 x+3)+(x-5) $$

Choose the correct letter or letters. No pencil is needed, just think your way through these. Which radical is not a real number? a. \(\sqrt{3}\) b. \(-\sqrt{11} \quad\) c. \(\sqrt[3]{-10} \quad\) d. \(\sqrt{-10}\)

3 because 8 is a perfect cube. $$ 16 $$ # Write each integer as a product of two integers such that one of the factors is a perfect cube. For example, write 24 as 8 # 3 because 8 is a perfect cube. $$ 16 $$

Write in the form \(a+b i\). $$ i^{3}-i^{4} $$

Perform each indicated operation. See Sections 1.4 and 5.4 $$ 9 y^{2}-8 y^{2} $$

Choose the correct letter or letters. No pencil is needed, just think your way through these. Which radical(s) simplify to \(3 ?\) a. \(\sqrt{9}\) b. \(\sqrt{-9}\) c. \(\sqrt[3]{27}\) d. \(\sqrt[3]{-27}\)

3 because 8 is a perfect cube. $$ 56 $$ # Write each integer as a product of two integers such that one of the factors is a perfect cube. For example, write 24 as 8 # 3 because 8 is a perfect cube. $$ 56 $$

Write in the form \(a+b i\). $$ i^{8}-i^{7} $$

Perform each indicated operation. See Sections 1.4 and 5.4 $$ \left(9 y^{2}\right)\left(-8 y^{2}\right) $$

Choose the correct letter or letters. No pencil is needed, just think your way through these. Which radical(s) simplify to \(-3 ?\) a. \(\sqrt{9}\) b. \(\sqrt{-9}\) c. \(\sqrt[3]{27}\) d. \(\sqrt[3]{-27}\)

3 because 8 is a perfect cube. $$ 54 $$ # Write each integer as a product of two integers such that one of the factors is a perfect cube. For example, write 24 as 8 # 3 because 8 is a perfect cube. $$ 54 $$

Write in the form \(a+b i\). $$ i^{6}+i^{8} $$

Perform each indicated operation. See Sections 1.4 and 5.4 $$ -3(x+5) $$

Choose the correct letter or letters. No pencil is needed, just think your way through these. Which radical does not simplify to a whole number? a. \(\sqrt{64}\) b. \(\sqrt[3]{64}\) c. \(\sqrt{8}\) d. \(\sqrt[3]{8}\)

3 because 8 is a perfect cube. $$ 80 $$ # Write each integer as a product of two integers such that one of the factors is a perfect cube. For example, write 24 as 8 # 3 because 8 is a perfect cube. $$ 80 $$

Write in the form \(a+b i\). $$ i^{4}+i^{12} $$

Perform each indicated operation. See Sections 1.4 and 5.4 $$ -3+x+5 $$

For Exercises 107 through \(110,\) do not use a calculator. \(\sqrt{160}\) is closest to a. 10 b. 13 c. 20 d. 40

Choose the correct letter for each exercise. Letters will be used more than once. No pencil is needed. Just think about the meaning of each expression. \(A=2, B=-2, C=\) not a real number $$ 4^{1 / 2} $$

Write in the form \(a+b i\). $$ 2+\sqrt{-9} $$

Perform each indicated operation. See Sections 1.4 and 5.4 $$ (x-4)^{2} $$

For Exercises 107 through \(110,\) do not use a calculator. \(\sqrt{1000}\) is closest to a. 10 b. 30 c. 100 d. 500

Choose the correct letter for each exercise. Letters will be used more than once. No pencil is needed. Just think about the meaning of each expression. \(A=2, B=-2, C=\) not a real number $$ -4^{1 / 2} $$

Write in the form \(a+b i\). $$ 5-\sqrt{-16} $$

Perform each indicated operation. See Sections 1.4 and 5.4 $$ (2 x+1)^{2} $$

Choose the correct letter for each exercise. Letters will be used more than once. No pencil is needed. Just think about the meaning of each expression. \(A=2, B=-2, C=\) not a real number $$ (-4)^{1 / 2} $$

Write in the form \(a+b i\). $$ \frac{6+\sqrt{-18}}{3} $$

Answer true or false. Assume all radicals represent nonzero real numbers. $$ \sqrt[n]{a} \cdot \sqrt[n]{b}=\sqrt[n]{a b} $$

Choose the correct letter for each exercise. Letters will be used more than once. No pencil is needed. Just think about the meaning of each expression. \(A=2, B=-2, C=\) not a real number $$ 8^{1 / 3} $$