Chapter 9

Fill in the blanks. The expressions \(4^{1 / 2}\) and \((-8)^{-2 / 3}\) have _____ exponents.

Fill in the blanks. The ______ number \(i\) is defined as \(i=\sqrt{-1} .\) We call \(i^{25}\) a _____________ of i.

Fill in the blanks. In this section, we used the ________ rule for radicals in reverse: \(\sqrt[n]{a} \cdot \sqrt[n]{b}=\sqrt[n]{a b}\).

Fill in the blanks. Radical expressions such as \(\sqrt[3]{4}\) and \(6 \sqrt[3]{4}\) with the same index and the same radicand are called _____ radicals.

Fill in the blanks. In a right triangle, the side opposite the \(90^{\circ}\) angle is called the _______.

\(5 x^{2}\) is the______ root of \(25 x^{4}\) because \(\left(5 x^{2}\right)^{2}=25 x^{4}\). The _______ root of 216 is 6 because \(6^{3}=216\).

Fill in the blanks. We read \(16^{3 / 2}\) as " 16 to the three- _____ power."

Fill in the blanks. A _____________ number is any number that can be written in the form \(a+b i,\) where \(a\) and \(b\) are real numbers and \(i=\sqrt{-1}\)

Fill in the blanks. To multiply \(2 \sqrt{5}(3 \sqrt{8}+\sqrt{3}),\) use the ___________ property.

Fill in the blanks. An _________ right triangle is a right triangle with two legs of equal length.

Fill in the blanks. Numbers such as 1, 4, 9, 16, 25, and 36 are called perfect _____ . Numbers such as 1, 8, 27, 64, and 125 are called perfect _____ . Numbers such as 1, 16, 81, 256, and 625 are called perfect-fourth _____ .

Fill in the blanks. To solve a radical equation, we find all the values of the variable that make the equation__.

The symbol \(\sqrt{\quad}\) is called a _______ symbol or a _______ root symbol.

Fill in the blanks. We read \(27^{-1 / 3}\) as " 27 to the _____ one-third power."

Fill in the blanks. For the complex number \(2+5 i\), we call 2 the_____________ part and 5 the _________ part.

Fill in the blanks. To __________ the denominator of \(\frac{4}{\sqrt{5}},\) we multiply the fraction by \(\frac{\sqrt{5}}{\sqrt{5}}\).

Fill in the blanks. The _________ theorem states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the hypotenuse.

Fill in the blanks. The largest perfect-square _____ of 27 is 9. The largest _____ -cube factor of 16 is 8.

Fill in the blanks. When we square both sides of a radical equation, we say we are__both sides to the second power.

A radical symbol \(\sqrt{ }\) represents the ___________ or principal square root of a number.

Fill in the blanks. We read \(\left(-64 a^{5}\right)^{4 / 5}\) as "the quantity of \(-64 a^{5}\), _____ to the four-fifths power."

Fill in the blanks. \(6+3 i\) and \(6-3 i\) are called complex_____________

Fill in the blanks. The denominator of the fraction \(\frac{4}{\sqrt{5}}\) is an __________ number.

Fill in the blanks. An ____________ triangle has three sides of equal length and three \(60^{\circ}\) angles.

Fill in the blanks. To _____ \(\sqrt{24}\) means to write it as \(2 \sqrt{6}.\)

Fill in the blanks. When solving equations containing radicals, first we __ one radical expression on one side of the equation.

The number 4 has two square roots, \(-2\) and \(2 .\) When we speak of the square root of \(4,\) we mean only the ________ root of \(4,\) which is 2.

Fill in the blanks. In the radical expression \(\sqrt[4]{16 x^{8}}, 4\) is the _____, and \(16 x^{8}\) is the _____.

Fill in the blanks. a. \(i=\) b. \(i^{2}=\) c. \(i^{3}=\) d. \(i^{4}=\) e. In general, the powers of \(i\) cycle through ________ possible outcomes.

Fill in the blanks. To obtain a ____________ -cube radicand in the denominator of \(\frac{\sqrt[3]{7}}{\sqrt[3]{5 n}}\) we multiply the fraction by \(\frac{\sqrt[3]{25 n^{2}}}{\sqrt[3]{25 n^{2}}}\).

Fill in the blanks. If \(a\) and \(b\) are the lengths of the legs of a right triangle and \(c\) is the length of the hypotenuse, then ___ + ___ = ____ This is called the Pythagorean _____.

Fill in the blanks. The product rule for radicals: \(\sqrt[n]{a b}=\quad\) In words, the nth root of the ____ of two numbers is equal to the product of their \(nth\) ____ .

Fill in the blanks. Proposed solutions of a radical cquation that don't satisfy it are called __ solutions.

The number 100 has two square roots. The positive ____________ or square root of 100 is \(10 .\)

Fill in the blanks. \(32^{4 / 5}\) means the fourth _____ of the fifth _____ of 32.

Fill in the blanks. Simplify: $$ \sqrt{-36}=\sqrt{\cdot 36}=\sqrt{36}=6 $$

Fill in the blanks. The __________ of \(\sqrt{x}+1\) is \(\sqrt{x}-1\).

Fill in the blanks. In any right triangle, the square of the hypotenuse is equal to the _____ of the squares of the two ____.

Fill in the blanks. The quotient rule for radicals: \(\sqrt[n]{\frac{a}{b}}=\). In words, the root of the ____ of two numbers is equal to the quotient of their \(nth\) _____.

Fill in the blanks. To __ a proposed solution means to substitute it into the original equation and see whether a true statement results.

In the expression \(\sqrt[3]{27 x^{6}},\) the \(\quad\) is 3 and \(27 x^{6}\) is the ______________.

Tell why each of the following expressions is not in simplified radical form. Then simplify it. Finally, use a calculator to approximate its value. $$ \begin{array}{|l|l|l|l|} \hline & \begin{array}{l} \text { Why isn't it in } \\ \text { simplified form? } \end{array} & \begin{array}{l} \text { Simplified } \\ \text { form } \end{array} & \text { Approximation } \\ \hline \frac{3}{\sqrt{2}} & & & \\ \hline \frac{\sqrt{18}}{2} & & & \\ \hline \sqrt{\frac{9}{2}} & & & \\ \hline \end{array} $$

Consider the expressions \(\sqrt{4 \cdot 5}\) and \(\sqrt{4} \sqrt{5} .\) Which expression is a. the square root of a product? b. the product of square roots? c. How are these two expressions related?

Fill in the blanks. In an isosceles right triangle, the length of the hypotenuse is ____ times the length of one leg.

Fill in the blanks. a. The power rule for solving radical equations states that if \(x\), \(y,\) and \(n\) are real numbers and \(x=y,\) then\(x=y\) b. If \(\sqrt[n]{a}\) is a real number, then \((\sqrt[n]{a})^{n}=\)

When we write \(\sqrt{b^{4}}=b^{2},\) we say that we have _____________ the radical expression.

Consider \(\frac{\sqrt[3]{a}}{\sqrt[3]{x^{2}}}\) and \(\sqrt[3]{\frac{a}{x^{2}}} .\) Which expression is a. the cube root of a quotient? b. the quotient of cube roots? c. How are these two expressions related?

Fill in the blank: To rationalize the denominator of \(\frac{3}{\sqrt{2}},\) we multiply it by \(\frac{\sqrt{2}}{\sqrt{2}},\) which is a form of __________.

The shorter leg of a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle is ____ as long as the hypotenuse.

Determine whether 6 is a solution of each radical equation. a. \(\sqrt{x+3}=x-3\) b. \(\quad \sqrt[3]{5 x-3}+9=x\)