Chapter 10

Complete sentence with the best choice from the column on the right. In any ____ triangle, the square of the length of the hypotenuse is the sum of the squares of the lengths of the legs. a) Hypotenuse b) Isosceles c) Legs d) Right e) Square roots f) \(30^{\circ}-60^{\circ}-90^{\circ}\)

In each of Exercises 1–8, match the expression with the equivalent expression from the column on the right. $$ x^{2 / 5} $$ a) \(x^{3 / 5}\) b) \((\sqrt[5]{x})^{4}\) c) \(\sqrt{x^{5}}\) d) \(x^{1 / 2}\) e)\(\frac{1}{(\sqrt{x})^{5}}\) f) \(\sqrt[4]{x^{3}}\) g) \(\sqrt[5]{x^{2}}\) h)\(\frac{1}{(\sqrt[5]{x})^{2}}\)

Complete sentence with the best choice from the column on the right. The shortest side of a right triangle is always one of the two _____. a) Hypotenuse b) Isosceles c) Legs d) Right e) Square roots f) \(30^{\circ}-60^{\circ}-90^{\circ}\)

Classify each of the following statements as either true or false. If \(t=7,\) then \(t^{2}=49\).

Fill in the blanks by selecting from the following words (which may be used more than once): radicand(s), indices, conjugate(s), base(s) denominator(s), numerator(s). To find a product by adding exponents, the _____ must be the same.

Classify each of the following statements as either true or false. Every imaginary number is a complex number, but not every complex number is imaginary.

Fill in the blanks by selecting from the following words (which may be used more than once): radicand(s), indices, conjugate(s), base(s) denominator(s), numerator(s). To add rational expressions, the ____ must be the same.

Classify each of the following statements as either true or false. Every real number is a complex number, but not every complex number is real.

Complete sentence with the best choice from the column on the right. In \(\mathrm{a}(\mathrm{n})\) ______ right triangle, both legs have the same length. a) Hypotenuse b) Isosceles c) Legs d) Right e) Square roots f) \(30^{\circ}-60^{\circ}-90^{\circ}\)

Concept Reinforcement In each of Exercises \(1-8\), match the expression with an equivalent expression from the column on the right. Assume \(a, b>0\) a) \(\frac{\sqrt[5]{a^{2}} \sqrt[5]{b^{2}}}{\sqrt[5]{b^{5}}}\) b \(\frac{a^{2}}{b^{3}}\) c) \(\sqrt{\frac{a \cdot b}{b^{3} \cdot b}}\) d) \(\sqrt{a}\) e) \(\frac{\sqrt[3]{a^{2}}}{b^{2}}\) f) \(\sqrt[5]{\frac{a^{6} b}{b^{4} \cdot b}}\) g) \(2 a\) h) \(\frac{\sqrt[5]{a^{2} b^{3}}}{\sqrt[5]{b^{5}}}\) $$ -\sqrt{\frac{a}{b^{3}}} $$

Fill in the blanks by selecting from the following words (which may be used more than once): radicand(s), indices, conjugate(s), base(s) denominator(s), numerator(s). To rationalize the ______ of \(\frac{\sqrt{c}-\sqrt{a}}{5},\) we multiply by a form of \(1,\) using the _____ of \(\sqrt{c}-\sqrt{a},\) or \(\sqrt{c}+\sqrt{a}\) to write 1

Classify each of the following statements as either true or false. $$\sqrt{x}-8=7 \text { is equivalent to } \sqrt{x}=15$$

Complete sentence with the best choice from the column on the right. In a(n) _____ right triangle, the hypotenuse is twice as long as the shorter leg. a) Hypotenuse b) Isosceles c) Legs d) Right e) Square roots f) \(30^{\circ}-60^{\circ}-90^{\circ}\)

Concept Reinforcement Classify each of the following statements as either true or false. The expression \(\sqrt[3]{X}\) is not simplified if \(X\) contains a factor that is a perfect cube.

Select the appropriate word to complete each of the following. If \(a\) is a whole number that is not a perfect square, then \(\sqrt{a}\) is a(n) ______ number.

Classify each of the following statements as either true or false. The product of a complex number and its conjugate is always a real number.

Complete sentence with the best choice from the column on the right. If both legs in a right triangle have measure \(a\), then the _____ measures \(a \sqrt{2}\). a) Hypotenuse b) Isosceles c) Legs d) Right e) Square roots f) \(30^{\circ}-60^{\circ}-90^{\circ}\)

Concept Reinforcement Classify each of the following statements as either true or false. It is often possible to simplify \(\sqrt{A \cdot B}\) even though \(\sqrt{A}\) and \(\sqrt{B}\) cannot be simplified.

Select the appropriate word to complete each of the following. The domain of the function \(f\) given by \(f(x)=\sqrt[3]{x}\) is the set of all ______ numbers..

Add or subtract. Simplify by combining like radical terms, if possible. Assume that all variables and radicands represent positive real numbers. $$2 \sqrt{5}+7 \sqrt{5}$$

Solve. $$\sqrt{5 x+1}=4$$

Multiply. $$ \sqrt{5} \sqrt{7} $$

Add or subtract. Simplify by combining like radical terms, if possible. Assume that all variables and radicands represent positive real numbers. $$4 \sqrt{7}+2 \sqrt{7} $$

Classify each of the following statements as either true or false. The quotient of two complex numbers is always a complex number.

Solve. $$\sqrt{7 x-3}=5$$

Multiply. $$ \sqrt{10} \sqrt{3} $$

Select the appropriate word to complete each of the following. If \(\sqrt[3]{x}\) is negative, then \(x\) must be _____.

Add or subtract. Simplify by combining like radical terms, if possible. Assume that all variables and radicands represent positive real numbers. $$7 \sqrt[3]{4}-5 \sqrt[3]{4}$$

Express in terms of \(i\) $$ \sqrt{-100} $$

Solve. $$\sqrt{3 x}+1=6$$

Multiply. $$ \sqrt[3]{3} \sqrt[3]{2} $$

Simplify by taking the roots of the numerator and the denominator. Assume that all variables represent positive numbers. $$ \sqrt{\frac{36}{25}} $$

Assume for all exercises that even roots are of non- negative quantities and that all denominators are nonzero. Write an equivalent expression using radical notation and, if possible, simplify. $$ x^{1 / 6} $$

For each number, find all of its square roots. $$ 49 $$

Add or subtract. Simplify by combining like radical terms, if possible. Assume that all variables and radicands represent positive real numbers. $$14 \sqrt[5]{2}-6 \sqrt[5]{2}$$

Express in terms of \(i\) $$ \sqrt{-25} $$

Solve. $$\sqrt{2 x}-1=2$$

Multiply. $$ \sqrt[3]{2} \sqrt[3]{5} $$

Simplify by taking the roots of the numerator and the denominator. Assume that all variables represent positive numbers. $$ \sqrt{\frac{100}{81}} $$

Assume for all exercises that even roots are of non- negative quantities and that all denominators are nonzero. Write an equivalent expression using radical notation and, if possible, simplify. $$ y^{1 / 5} $$

For each number, find all of its square roots. $$ 81 $$

Add or subtract. Simplify by combining like radical terms, if possible. Assume that all variables and radicands represent positive real numbers. $$\sqrt[3]{y}+9 \sqrt[3]{y}$$

Express in terms of \(i\) $$ \sqrt{-13} $$

Solve. $$\sqrt{y+1}-5=8$$

Simplify by taking the roots of the numerator and the denominator. Assume that all variables represent positive numbers. $$ \sqrt[3]{\frac{64}{27}} $$

Assume for all exercises that even roots are of non- negative quantities and that all denominators are nonzero. Write an equivalent expression using radical notation and, if possible, simplify. $$ 16^{1 / 2} $$

For each number, find all of its square roots. $$ 144 $$

Add or subtract. Simplify by combining like radical terms, if possible. Assume that all variables and radicands represent positive real numbers. $$9 \sqrt[4]{t}-\sqrt[4]{t}$$

Express in terms of \(i\) $$ \sqrt{-19} $$

Solve. $$\sqrt{x-2}-7=-4$$