Chapter 3

Brandon said that if \(a\) is a positive real number, then \(3 a, 4 a,\) and 5\(a\) are the lengths of the sides of a right triangle. Therefore, \(3 \sqrt{2}, 4 \sqrt{2},\) and 5\(\sqrt{2}\) are the lengths of the sides of a right triangle. Do you agree with Brandon? Justify your answer.

Explain why \(\sqrt{x+3}<0\) has no solution in the set of real numbers while \(\sqrt{x+3} \geq 0\) is true for all real numbers greater than or equal to \(-3 .\)

Justin simplified \(\frac{7}{2 \sqrt{7}}\) by first writing 7 as \(\sqrt{49}\) and then dividing numerator and denominator by \(\sqrt{7}\). a. Show that Justin's solution is correct. b. Can \(\frac{7}{2 \sqrt{5}}\) be simplified by using the same procedure? Explain why or why not.

a. Kevin said that if the index of a radical is even and the radicand is positive, then the radical has two real roots. Do you agree with Kevin? Explain why or why not. b. Kevin said that if the index of a radical is odd, then the radical has one real root and that the root is positive if the radicand is positive and negative if the radicand is negative. Do you agree with Kevin? Explain why or why not.

Jonathan said that \(\frac{\sqrt{10}}{2}=\sqrt{5} .\) Do you agree with Jonathan? Justify your answer.

Danielle said that 3\(x \sqrt{\frac{1}{3 x}}\) could be simplified by writing 3\(x \sqrt{\frac{1}{3 x}}\) as \(\sqrt{\frac{9 x^{2}}{3 x}}=\sqrt{3 x} .\) Do you agree with Danielle? Justify your answer.

Explain the difference between \(-\sqrt{36}\) and \(\sqrt{-36}\)

Tony said that \(\frac{3}{1-\frac{1}{5}}\) is irrational because it is not the ratio of integers and is therefore not a rational number. Do you agree with Tony? Explain why or why not.

Jennifer said that if \(a\) is a positive real number, then \(\sqrt[4]{a^{2}}=\sqrt{a} .\) Do you agree with Jennifer? Justify your answer.

To rationalize the denominator of \(\frac{4}{2+\sqrt{8}},\) Brittany multiplied by \(\frac{2-\sqrt{8}}{2}\) and Justin multiplied by \(\frac{1-\sqrt{2}}{1-\sqrt{2}} .\) Explain why both are correct.

a. Sarah said that in the set of real numbers, \(\sqrt{a}\) is one of the two equal factors whose product is \(a .\) Therefore, \(\sqrt{a} \cdot \sqrt{a}=a\) for some values of \(a\) . Do you agree with Sarah? Explain why or why not. b. If you agree with Sarah, for which values of \(a\) is the statement true? Explain.

Show that the quotient of two irrational numbers can be either rational or irrational.

Does \(\sqrt{16}+\sqrt{48}=\sqrt{64} ?\) Justify your answer.

If \(a\) is a negative number, is \(-\sqrt[3]{-8 a^{3}}\) a positive number, a negative number, or not a real number? Explain your answer. oning Skills.

Maria said that since the solution of the inequality \(|2 x-5| < 3\) can be found by using \(-3 < 2 x-5 < 3,\) then the solution of the inequality \(|2 x-5| > 3\) can be found by using \(-3 > 2 x-5 > 3 .\) Do you agree with Maria? Explain why or why not.

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ \sqrt{2} \cdot \sqrt{8} $$

In \(3-38,\) solve each equation for the variable, check, and write the solution set. $$ \sqrt{a}=5 $$

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{1}{\sqrt{3}}\)

In \(3-10,\) tell whether each represents a number that is rational, irrational, or neither. $$ \sqrt{25} $$

In \(3-29\) write each quotient in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{24} \div \sqrt{6} $$

In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{2}+5 \sqrt{2} $$

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{12} $$

In \(3-14,\) determine whether each of the numbers is rational or irrational. $$ \frac{0}{4} $$

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ \sqrt{5} \cdot \sqrt{45} $$

In \(3-38,\) solve each equation for the variable, check, and write the solution set. $$ \sqrt{x}=7 $$

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{5}{\sqrt{10}}\)

In \(3-10,\) tell whether each represents a number that is rational, irrational, or neither. $$ \sqrt{8} $$

In \(3-29\) write each quotient in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{75} \div \sqrt{3} $$

In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ 6 \sqrt{5}-4 \sqrt{5} $$

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{50} $$

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ \sqrt{3} \cdot \sqrt{27} $$

In \(3-38,\) solve each equation for the variable, check, and write the solution set. $$ 4 \sqrt{y}=12 $$

In \(3-10,\) tell whether each represents a number that is rational, irrational, or neither. $$ \sqrt{-8} $$

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{4}{\sqrt{2}}\)

In \(3-29\) write each quotient in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{72} \div \sqrt{8} $$

In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ 8 \sqrt{3}+\sqrt{3} $$

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{32} $$

In \(3-14,\) determine whether each of the numbers is rational or irrational. $$ 3 \pi $$

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ \sqrt{8} \cdot \sqrt{12} $$

In \(3-38,\) solve each equation for the variable, check, and write the solution set. $$ \sqrt{4 y}=12 $$

In \(3-10,\) tell whether each represents a number that is rational, irrational, or neither. $$ \sqrt[3]{-8} $$

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{4}{2 \sqrt{3}}\)

In \(3-29\) write each quotient in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{50 a^{3}} \div \sqrt{5 a} $$

In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ 5 \sqrt{7}-\sqrt{7} $$

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt[4]{48 a^{9} b^{3}} $$

In \(3-14,\) determine whether each of the numbers is rational or irrational. $$ \sqrt{17} $$

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ -\sqrt{10} \cdot \sqrt{18} $$

In \(3-38,\) solve each equation for the variable, check, and write the solution set. $$ 2 \sqrt{b}=8 $$

In \(3-10,\) tell whether each represents a number that is rational, irrational, or neither. $$ \sqrt{0} $$

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{15}{5 \sqrt{3}}\)