Chapter 7

Fill in the blanks. \(5 x^{2}\) 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. The ____ number \(i\) is defined as \(i=\sqrt{-1} .\) We call \(i^{25}\) a _____ of \(i.\)

The expressions \(4^{1 / 2}\) and \((-8)^{-2 / 3}\) have __________ exponents.

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

Fill in the blanks. Equations such as \(\sqrt{x+4}-4=5\) and \(\sqrt[3]{x+1}=12\) are called _____ equations.

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

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 \(i=\sqrt{-1}.\)

We read \(16^{3 / 2}\) as " 16 to the three-_______ power.”

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

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

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 multiply \(2 \sqrt{5}(3 \sqrt{8}+\sqrt{3}),\) use the _____ property.

Fill in the blanks. A radical symbol \(\sqrt{\quad}\) represents the _____ or principal square root of a number.

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

We read \(27^{-1 / 3}\) as " 27 to the _______one-third power.”

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.

To ____ the denominator of \(\frac{4}{\sqrt{5}},\) we multiply the fraction by \(\frac{\sqrt{5}}{\sqrt{5}}\)

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

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

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.

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

Fill in the blanks. The number 100 has two square roots. The positive or _____ square root of 100 is 10.

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

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. If \(a\) and \(b\) are the lengths of the legs of a right triangle and \(c\) is the length of the hypotenuse, then \(\square\)+\(\square$$=\) This is called the Pythagorean _____.

Fill in the blanks. The product rule for radicals: \(\sqrt[n]{a b}= In words, the \)n\( th root of the _of two numbers is equal to the product of their \)n t h$ ___.

Fill in the blanks. Proposed solutions of a radical equation that don’t satisfy it are called _____ solutions.

Fill in the blanks. In the expression \(\sqrt[3]{27 x^{6}},\) the _____ is 3 and \(27 x^{6}\) is the _____.

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

\(32^{4 / 5}\) means the fourth ________ of the fifth _________of 32.

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: \(\sqrt[n]{\frac{a}{b}}=\quad .\) In words, the \(n\) th root of the __ of two numbers is equal to the quotient of their \(n\) th ___.

To _____ a proposed solution means to substitute it into the original equation and see whether a true statement results.

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

Fill in the blanks. a. To add (or subtract) complex numbers, add (or subtract) their _____ parts. b. To multiply two complex numbers, such as \((2+3 i)(3+5 i),\) we can use the _____ method.

Complete the table by writing the given expression in the alternate form. Also give the base and exponent for the exponential form. \begin{tabular}{|c|c|c|c|} \hline Radical form & Exponential form & Base & Exponent \\ \hline\(\sqrt[5]{25}\) & & & \\ \((\sqrt[4]{16})^{-3}\) & \((-27)^{2 / 3}\) & & \\ & & \\ \(-\sqrt{\frac{9}{64}}\) & & & \\ \hline \end{tabular}

Fill in the blanks. In an isosceles right triangle, the length of the hypotenuse is \(\square\) 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 \((\sqrt[n]{a})^{n}=\)

Fill in the blanks. 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?

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

Fill in the blanks. When \(n\) is an odd number, \(\sqrt[n]{x}\) represents an _____ root. When \(n\) is an _____ number, \(\sqrt[n]{x}\) represents an even root.

Fill in the blanks. To divide \(6+7 i\) by \(1-8 i,\) we multiply \(\frac{6+7 i}{1-8 i}\) by 1 in the form of _____ .

Fill in the blanks. 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. \(\sqrt[3]{5 x-3}+9=x\)

Fill in the blanks. 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?