Rational and Radical Expressions

Rational Expressions

A rational expression is one which can be written as the quotient of two polynomials. (Hence any polynomial is also a rational expression.) Rational expressions are defined for all real values of the variables except those that make the denominator equal to zero.

EXAMPLE 4.1 $ , (y ) $ ; $ , (x ) $ ; $ y^{3} - 5y^{2} $ ; $ x^{3} - 3x^{2} + 8x - $ , $ (x ) $ are examples of rational expressions.

Fundamental Principle of Fractions

For all real numbers \(a, b, k (b, k \neq 0)\)

\[ \frac{a}{b}=\frac{ak}{bk}\left(building to higher terms\right)\qquad\frac{ak}{bk}=\frac{a}{b}\left(reducing to lower terms\right) \]

EXAMPLE 4.2 Reducing to lowest terms: $ == $

Operations on Rational Expressions

Operations on rational expressions (all denominators assumed $ $ ):

\[ \begin{aligned}&\left(\frac{a}{b}\right)^{-1}=\frac{b}{a}\quad&\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}\quad&\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\left(\frac{c}{d}\right)^{-1}=\frac{a}{b}\cdot\frac{d}{c}=\frac{ad}{bc}\\&\frac{a}{c}\pm\frac{b}{c}=\frac{a\pm b}{c}\quad&\frac{a}{b}\pm\frac{c}{d}=\frac{ad}{bd}\pm\frac{bc}{bd}=\frac{ad\pm bc}{bd}\end{aligned} \]

Note: In addition to expressions with unequal denominators, the result is usually written in lowest terms, and the expressions are built to higher terms using the lowest common denominator (LCD).

EXAMPLE 4.3 Subtraction: $ - = - = $

Complex Fractions

Complex fractions are expressions containing fractions in the numerator and/or denominator. They can be reduced to simple fractions by two methods:

Method 1: Combine numerator and denominator into single quotients, then divide.

EXAMPLE 4.4

\[ \begin{aligned}\frac{\frac{x}{x-1}-\frac{a}{a-1}}{x-a}&=\frac{\frac{x(a-1)-a(x-1)}{(x-1)(a-1)}}{x-a}=\frac{xa-x-ax+a}{(x-1)(a-1)}\div(x-a)\\&=\frac{a-x}{(x-1)(a-1)}\cdot\frac{1}{x-a}=\frac{-1}{(x-1)(a-1)}\end{aligned} \]

Method 2: Multiply numerator and denominator by the LCD of all internal fractions:

EXAMPLE 4.5

\[ \frac{\frac{x}{y}-\frac{y}{x}}{\frac{x}{y^{2}}+\frac{y}{x^{2}}}=\frac{\frac{x}{y}-\frac{y}{x}}{\frac{x}{y^{2}}+\frac{y}{x^{2}}}\cdot\frac{x^{2}y^{2}}{x^{2}y^{2}}=\frac{x^{3}y-xy^{3}}{x^{3}+y^{3}}=\frac{xy(x-y)(x+y)}{(x+y)(x^{2}-xy+y^{2})}=\frac{xy(x-y)}{x^{2}-xy+y^{2}} \]

Method 2 is more convenient when the fractions in numerator and denominator involve very similar expressions.

Rational Expressions

Rational expressions are often written in terms of negative exponents.

EXAMPLE 4.6 Simplify: $ x^{-3}y{5} - 3x^{-4}y{6} $

This can be done in two ways, either by removing the common factor of $ x^{-4}y{5} $ , as in the previous chapter, or by rewriting as the sum of two rational expressions:

\[ x^{-3}y^{5}-3x^{-4}y^{6}=\frac{y^{5}}{x^{3}}-\frac{3y^{6}}{x^{4}}=\frac{xy^{5}}{x^{4}}-\frac{3y^{6}}{x^{4}}=\frac{xy^{5}-3y^{6}}{x^{4}}. \]

Radical Expressions

For n a natural number greater than 1 and x a real number, the nth root radical is defined to be the principal nth root of x:

\[ \sqrt[n]{x}=x^{1/n} \]

If n = 2, write $ $ in place of $ $ .

The symbol ✓ is called a radical, n is called the index, and x is called the radicand.

Properties of Radicals

\[ (\bigvee^{n}\overline{x})^{n}=x,if\bigvee^{n}\overline{x}is defined \]

\[ \sqrt[n]{x^{n}}=x,if x\geq0 \]

\[ \sqrt[n]{x^{n}}=x,if x<0,n odd \]

\[ \bigvee^{n}\sqrt{x^{n}}=\left|x\right|,if x<0,n even \]

\[ \bigvee^{n}\sqrt{ab}=\bigvee^{n}\sqrt{a}\bigvee^{n}\sqrt{b} \]

\[ \sqrt[n]{\sqrt[m]{x}}=\sqrt[mn]{x} \]

\[ \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}} \]

Unless otherwise specified, it is normally assumed that variable bases represent nonnegative real numbers.

Simplest Radical Form

Simplest radical form for radical expressions:

1. No radicand can contain a factor with an exponent greater than or equal to the index of the radical.

2. No power of the radicand and the index of the radical can have a common factor other than 1.

3. No radical appears in a denominator.

4. No fraction appears in a radical.

EXAMPLE 4.7

1. $ $ violates condition 1. It is simplified as follows:

\[ \sqrt[3]{16x^{3}y^{5}}=\sqrt[3]{8x^{3}y^{3}\cdot2y^{2}}=\sqrt[3]{8x^{3}y^{3}}\cdot\sqrt[3]{2y^{2}}=2xy\sqrt[3]{2y^{2}} \]

2. $ $ violates condition 2. It is simplified as follows:

\[ \sqrt[6]{t^{3}}=\sqrt[2\cdot3]{t^{3}}=\sqrt{\sqrt[3]{t^{3}}}=\sqrt{t} \]

3. $ $ violates condition 3. It is simplified as follows:

\[ \frac{12x^{2}}{\sqrt[4]{27xy^{2}}}=\frac{12x^{2}}{\sqrt[4]{27xy^{2}}}\cdot\frac{\sqrt[4]{3x^{3}y^{2}}}{\sqrt[4]{3x^{3}y^{2}}}=\frac{12x^{2}\sqrt[4]{3x^{3}y^{2}}}{\sqrt[4]{81x^{4}y^{4}}}=\frac{12x^{2}\sqrt[4]{3x^{3}y^{2}}}{3xy}=\frac{4x\sqrt[4]{3x^{3}y^{2}}}{y} \]

4. $ $ violates condition 4. It is simplified as follows:

\[ \sqrt[4]{\frac{3x}{5y^{3}}}=\sqrt[4]{\frac{3x}{5y^{3}}\cdot\frac{5^{3}y}{5^{3}y}}=\sqrt[4]{\frac{375xy}{5^{4}y^{4}}}=\frac{\sqrt[4]{375xy}}{5y} \]

Satisfying condition 3 is often referred to as rationalizing the denominator.

The conjugate expression for a binomial of form $ a + b $ is the expression a - b and conversely.

EXAMPLE 4.8 Rationalize the denominator: $ $

Multiply numerator and denominator by the conjugate expression for the denominator:

\[ \frac{x-4}{\sqrt{x}-2}=\frac{x-4}{\sqrt{x}-2}\cdot\frac{\sqrt{x}+2}{\sqrt{x}+2}=\frac{(x-4)(\sqrt{x}+2)}{x-4}=\sqrt{x}+2 \]

Expressions are not always written in simplest radical form. Often it is important to rationalize the numerator.

EXAMPLE 4.9 Rationalize the numerator: $ $

Multiply numerator and denominator by the conjugate expression for the numerator.

\[ \frac{\sqrt{x}-\sqrt{a}}{x-a}=\frac{\sqrt{x}-\sqrt{a}}{x-a}\cdot\frac{\sqrt{x}+\sqrt{a}}{\sqrt{x}+\sqrt{a}}=\frac{x-a}{(x-a)\left(\sqrt{x}+\sqrt{a}\right)}=\frac{1}{\sqrt{x}+\sqrt{a}} \]

Conversion of Radical Expressions

Conversion of radical expressions to exponent form:

For m, n positive integers (n > 1) and $ x $ when n is even,

\[ \sqrt[n]{x^{m}}=x^{m/n} \]

Conversely, $ x^{m/n} = $

\[ \mathrm{A l s o},x^{m/n}=(\sqrt[n]{x})^{m} \]

EXAMPLE 4.10 (a) \(\sqrt[3]{x}=x^{1/3}\); (b) \(\sqrt[4]{x^{3}}=x^{3/4}\); (c) \(\sqrt[5]{x^{5}}=x^{5/2}\)

Operations with Complex Numbers

Complex numbers can be written in standard form $ a + bi $ . In this form, they can be combined using the operations defined for real numbers, together with the definition of the imaginary unit i: $ i^{2} = -1 $ . The conjugate of a complex number z is denoted $ {z} $ . If $ z = a + bi $ , then $ {z} = a - bi $

EXAMPLE 4.11 (a) Write 4 − √−25 in standard form. (b) Find the conjugate of 3 − 7i. (c) Simplify (3 + 4i)2.

1. $ 4 - = 4 - = 4 - 5i $

2. The conjugate of 3 - 7i is $ 3 - (-7i) $ or $ 3 + 7i $ .

SOLVED PROBLEMS

4.1. Reduce to lowest terms: (a) $ $ (b) $ $ (c) $ $

1. First factor numerator and denominator, then reduce by removing common factors:

\[ \frac{5x^{2}-8x+3}{25x^{2}-9}=\frac{(5x-3)(x-1)}{(5x-3)(5x+3)}=\frac{x-1}{5x+3} \]

\[ \frac{x^{3}-a^{3}}{x-a}=\frac{(x-a)(x^{2}+ax+a^{2})}{x-a}=x^{2}+ax+a^{2} \]

4.2. Explain why every polynomial is also a rational expression.

A rational expression is one which can be written as the quotient of two polynomials. Every polynomial P can be written as P/1, where numerator and denominator are polynomials; hence every polynomial is also a rational expression.

4.3. Perform indicated operations:

1. $ $ (b) $ (x^{2}-3xy+2y{2}) $ (c) $ - $

2. $ +- $

3. Factor all numerators and denominators, then reduce by removing any common factors.

\[ \frac{x^{2}-7x+12}{x^{2}-9}\cdot\frac{x^{3}-6x^{2}+9x}{x^{3}-4x^{2}}=\frac{(x-3)(x-4)}{(x-3)(x+3)}\cdot\frac{x(x-3)^{2}}{x^{2}(x-4)}=\frac{(x-3)^{2}}{x(x+3)} \]

2. Change division to multiplication, then proceed as in (a).

\[ \begin{aligned}\frac{x^{2}-4y^{2}}{xy+2y^{2}}\div(x^{2}-3xy+2y^{2})&=\frac{x^{2}-4y^{2}}{xy+2y^{2}}\cdot\frac{1}{x^{2}-3xy+2y^{2}}=\frac{(x-2y)(x+2y)}{y(x+2y)}\cdot\frac{1}{(x-y)(x-2y)}\\&=\frac{1}{y(x-y)}\end{aligned} \]

3. Find the lowest common denominator, then build to higher terms and perform the subtraction.

\[ \frac{1}{x+h}-\frac{1}{x}=\frac{x}{x(x+h)}-\frac{(x+h)}{x(x+h)}=\frac{x-(x+h)}{x(x+h)}=\frac{-h}{x(x+h)} \]

4. Proceed as in (c).

\[ \begin{aligned}\frac{2}{x-1}+\frac{3}{x+1}-\frac{4x-2}{x^{2}-1}&=\frac{2(x+1)}{(x-1)(x+1)}+\frac{3(x-1)}{(x-1)(x+1)}-\frac{(4x-2)}{(x-1)(x+1)}\\&=\frac{2(x+1)+3(x-1)-(4x-2)}{(x-1)(x+1)}=\frac{2x+2+3x-3-4x+2}{(x-1)(x+1)}\\&=\frac{x+1}{(x-1)(x+1)}=\frac{1}{x-1}\end{aligned} \]

4.4. Write each complex fraction as a simple fraction in lowest terms:

1. $ $ (b) $ $ (c) $ $ (d) $ $

2. Multiply numerator and denominator by y, the only internal denominator:

\[ \frac{y+\frac{x^{2}}{y}}{y^{2}}=\frac{y+\frac{x^{2}}{y}}{y^{2}}\cdot\frac{y}{y}=\frac{y^{2}+x^{2}}{y^{3}} \]

2. Multiply numerator and denominator by $ (x - 1)(x + 1) $ , the LCD of the internal fractions:

\[ \frac{\frac{x}{x-1}-\frac{x}{x+1}}{\frac{x}{x-1}+\frac{x}{x+1}}=\frac{\frac{x}{x-1}-\frac{x}{x+1}}{\frac{x}{x-1}+\frac{x}{x+1}}\cdot\frac{(x-1)(x+1)}{(x-1)(x+1)}=\frac{x(x+1)-x(x-1)}{x(x+1)+x(x-1)}=\frac{x^{2}+x-x^{2}+x}{x^{2}+x+x^{2}-x}=\frac{2x}{2x^{2}}=\frac{1}{x} \]

3. Combine numerator and denominator into single quotients, then divide:

\[ \begin{aligned}\frac{\frac{2}{x+2}-3}{\frac{4}{x}-x}&=\frac{\frac{2}{x+2}-\frac{3(x+2)}{x+2}}{\frac{4}{x}-\frac{x^{2}}{x}}=\frac{\frac{2-3x-6}{x+2}}{\frac{4-x^{2}}{x}}=\frac{-3x-4}{x+2}\div\frac{4-x^{2}}{x}\\&=\frac{-3x-4}{x+2}\cdot\frac{x}{4-x^{2}}=\frac{-3x^{2}-4x}{(x+2)(4-x^{2})}\end{aligned} \]

4. Proceed as in (c):

\[ \frac{1}{x-\ a}=\frac{\frac{a}{ax}-\frac{x}{ax}}{x-a}=\frac{\frac{a-x}{ax}}{x-a}=\frac{a-x}{ax}\div(x-a)=\frac{-(x-a)}{ax}\cdot\frac{1}{x-a}=-\frac{1}{ax} \]

4.5. Simplify: (a) $ 3(x + 3)^{2}(2x - 1)^{-4} - 8(x + 3)^{3}(2x - 1)^{-5} $ (b) $ $

1. Remove the common factor $ (x + 3)^{2}(2x - 1)^{-5} $ first:

\[ \begin{aligned}3(x+3)^{2}(2x-1)^{-4}-8(x+3)^{3}(2x-1)^{-5}&=(x+3)^{2}(2x-1)^{-5}[3(2x-1)-8(x+3)]\\&=(x+3)^{2}(2x-1)^{-5}[-2x-27]\\&=-\frac{(x+3)^{2}(2x+27)}{(2x-1)^{5}}\\ \end{aligned} \]

2. Eliminate negative exponents, then multiply numerator and denominator by $ x^{2}(x + h)^{2} $ , the LCD of the internal denominators:

\[ \begin{aligned}\frac{(x+h)^{-2}-x^{-2}}{h}&=\frac{\frac{1}{(x+h)^{2}}-\frac{1}{x^{2}}}{h}\cdot\frac{x^{2}(x+h)^{2}}{x^{2}(x+h)^{2}}=\frac{x^{2}-(x+h)^{2}}{hx^{2}(x+h)^{2}}=\frac{x^{2}-x^{2}-2xh-h^{2}}{hx^{2}(x+h)^{2}}\\&=\frac{-2xh-h^{2}}{hx^{2}(x+h)^{2}}=\frac{h(-2x-h)}{hx^{2}(x+h)^{2}}=\frac{-2x-h}{x^{2}(x+h)^{2}}\end{aligned} \]

4.6. Write in simplest radical form:

1. $ $ (b) $ $ (c) $ $ (d) $ $

2. Remove the largest possible perfect square factor, then apply the rule \(\sqrt{ab} = \sqrt{a}\sqrt{b}\)

\[ \sqrt{20x^{3}y^{4}z^{5}}=\sqrt{4x^{2}y^{4}z^{4}}\cdot5xz=\sqrt{4x^{2}y^{4}z^{4}}\cdot\sqrt{5xz}=2xy^{2}z^{2}\sqrt{5xz} \]

2. Remove the largest possible perfect cube factor, then apply the rule $ = $ .

\[ \sqrt[3]{108x^{5}(x+y)^{6}}=\sqrt[3]{27x^{3}(x+y)^{6}}\sqrt[3]{4x^{2}}=3x(x+y)^{2}\cdot\sqrt[3]{4x^{2}} \]

3. Build to higher terms so that the denominator is a perfect square, then apply $ = $ :

\[ \sqrt{\frac{3x}{5y}}=\sqrt{\frac{3x}{5y}\cdot\frac{5y}{5y}}=\sqrt{\frac{15xy}{25y^{2}}}=\frac{\sqrt{15xy}}{\sqrt{25y^{2}}}=\frac{\sqrt{15xy}}{5y} \]

4. Build to higher terms so that the denominator is a perfect cube, then apply $ = $ :

\[ \sqrt[3]{\frac{2x^{4}}{9yz^{2}}}=\sqrt[3]{\frac{2x^{4}}{9yz^{2}}\cdot\frac{3y^{2}z}{3y^{2}z}}=\sqrt[3]{\frac{6x^{4}y^{2}z}{27y^{3}z^{3}}}=\frac{\sqrt[3]{6x^{4}y^{2}z}}{\sqrt[3]{27y^{3}z^{3}}}=\frac{\sqrt[3]{x^{3}\cdot6xy^{2}z}}{3yz}=\frac{x\sqrt[3]{6xy^{2}z}}{3yz} \]

4.7. Rationalize the denominator:

1. $ $ (b) $ $ (c) $ $ (d) $ $

2. Build to higher terms so that the denominator becomes the cube root of a perfect cube, then reduce:

\[ \frac{x^{3}y^{2}}{\sqrt[3]{2xy^{2}}}=\frac{x^{3}y^{2}}{\sqrt[3]{2xy^{2}}}\cdot\frac{\sqrt[3]{4x^{2}y}}{\sqrt[3]{4x^{2}y}}=\frac{x^{3}y^{2}\sqrt[3]{4x^{2}y}}{\sqrt[3]{8x^{3}y^{3}}}=\frac{x^{3}y^{2}\sqrt[3]{4x^{2}y}}{2xy}=\frac{x^{2}y\ \sqrt[3]{4x^{2}y}}{2} \]

2. Build to higher terms using $ -1 $ , the conjugate expression for the denominator:

\[ \frac{\sqrt{x}}{\sqrt{x+1}}=\frac{\sqrt{x}}{\sqrt{x}+1}\cdot\frac{\sqrt{x}-1}{\sqrt{x}-1}=\frac{x-\sqrt{x}}{x-1} \]

3. Proceed as in (b):

\[ \frac{\sqrt{x}+\sqrt{h}}{\sqrt{x}-\sqrt{h}}=\frac{\sqrt{x}+\sqrt{h}}{\sqrt{x}-\sqrt{h}}\cdot\frac{\sqrt{x}+\sqrt{h}}{\sqrt{x}+\sqrt{h}}=\frac{(\sqrt{x}+\sqrt{h})^{2}}{x-h}=\frac{x+2\sqrt{x h}+h}{x-h} \]

4. Proceed as in (b):

\[ \frac{x^{2}-16y^{2}}{\sqrt{x}-2\sqrt{y}}=\frac{x^{2}-16y^{2}}{\sqrt{x}-2\sqrt{y}}\cdot\frac{\sqrt{x}+2\sqrt{y}}{\sqrt{x}+2\sqrt{y}}=\frac{(x^{2}-16y^{2})(\sqrt{x}+2\sqrt{y})}{x-4y}=(x+4y)(\sqrt{x}+2\sqrt{y}) \]

4.8. Rationalize the numerator:

1. $ $ (b) $ $ (c) $ $

2. Build to higher terms using $ $ :

\[ \frac{\sqrt{x}}{\sqrt{x}+1}=\frac{\sqrt{x}}{\sqrt{x}+1}\cdot\frac{\sqrt{x}}{\sqrt{x}}=\frac{x}{x+\sqrt{x}} \]

2. Build to higher terms using $ - $ , the conjugate expression for the numerator:

\[ \frac{\sqrt{x}+\sqrt{h}}{\sqrt{x}-\sqrt{h}}=\frac{\sqrt{x}+\sqrt{h}}{\sqrt{x}-\sqrt{h}}\cdot\frac{\sqrt{x}-\sqrt{h}}{\sqrt{x}-\sqrt{h}}=\frac{x-h}{(\sqrt{x}-\sqrt{h})^{2}}=\frac{x-h}{x-2\sqrt{x h}+h} \]

3. Proceed as in (b):

\[ \begin{aligned}\frac{\sqrt{x+h}-\sqrt{x}}{h}&=\frac{\sqrt{x+h}-\sqrt{x}}{h}\cdot\frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}=\frac{x+h-x}{h(\sqrt{x+h}+\sqrt{x})}\\&=\frac{h}{h(\sqrt{x+h}+\sqrt{x})}=\frac{1}{\sqrt{x+h}+\sqrt{x}}\end{aligned} \]

4.9. Write in exponent notation: (a) \(\sqrt{xy^{3}}\) (b) \(\sqrt[3]{a^{2}b(x-y)^{5}}\)

1. $ = (xy^{3}){1/2} = x^{1/2}y{3/2}; $ (b) $ = [a^{2}b(x - y)^{5}]^{1/3} = a^{2/3}b{1/3}(x - y)^{5/3} $

4.10. Write as a sum or difference of terms in exponential notation:

1. $ $ (b) $ $

2. $ ==-=x^{1/2}-x{-1/2} $

\[ \begin{aligned}(b)\frac{x^{3}-6x^{2}+3x+1}{6\sqrt[3]{x^{5}}}&=\frac{x^{3}-6x^{2}+3x+1}{6x^{5/3}}=\frac{x^{3}}{6x^{5/3}}-\frac{6x^{2}}{6x^{5/3}}+\frac{3x}{6x^{5/3}}+\frac{1}{6x^{5/3}}\\&=\frac{1}{6}x^{4/3}-x^{1/3}+\frac{1}{2}x^{-2/3}+\frac{1}{6}x^{-5/3}\end{aligned} \]

4.11. Write as a single fraction in lowest terms. Do not rationalize denominators.

\[ \begin{aligned}(a)\quad\sqrt{x-2}+\frac{2}{\sqrt{x-2}}\quad&(b)\frac{\sqrt{x^{2}-1}-\frac{x^{2}}{\sqrt{x^{2}-1}}}{x^{2}-1}\quad&(c)\frac{x^{2}(x^{2}+9)^{-1/2}-\sqrt{x^{2}+9}}{x^{2}}\end{aligned} \]

\[ \frac{(x^{2}-9)^{1/3}\ 3-(4x)\Big(\frac{1}{3}\Big)(x^{2}-9)^{-2/3}(2x)}{[(x^{2}-9)^{1/3}]^{2}} \]

\[ \begin{aligned}\sqrt{x-2}+\frac{2}{\sqrt{x-2}}&=\frac{\sqrt{x-2}}{1}+\frac{2}{\sqrt{x-2}}=\frac{\sqrt{x-2}\cdot\sqrt{x-2}}{\sqrt{x-2}}+\frac{2}{\sqrt{x-2}}=\frac{x-2+2}{\sqrt{x-2}}\\&=\frac{x}{\sqrt{x-2}}\end{aligned} \]

2. Multiply numerator and denominator by $ $ , the only internal denominator:

\[ \frac{\sqrt{x^{2}-1}-\frac{x^{2}}{\sqrt{x^{2}-1}}}{x^{2}-1}=\frac{\sqrt{x^{2}-1}-\frac{x^{2}}{\sqrt{x^{2}-1}}}{x^{2}-1}\cdot\frac{\sqrt{x^{2}-1}}{\sqrt{x^{2}-1}}=\frac{x^{2}-1-x^{2}}{(x^{2}-1)\sqrt{x^{2}-1}}=-\frac{1}{(x^{2}-1)^{3/2}} \]

3. Rewrite in exponent notation, then remove the common factor $ (x^{2} + 9)^{-1/2} $ from the numerator:

\[ \begin{aligned}\frac{x^{2}(x^{2}+9)^{-1/2}-\sqrt{x^{2}+9}}{x^{2}}&=\frac{x^{2}(x^{2}+9)^{-1/2}-(x^{2}+9)^{1/2}}{x^{2}}=\frac{(x^{2}+9)^{-1/2}[x^{2}-(x^{2}+9)^{1}]}{x^{2}}\\&=\frac{x^{2}-x^{2}-9}{(x^{2}+9)^{1/2}x^{2}}=\frac{-9}{x^{2}(x^{2}+9)^{1/2}}\end{aligned} \]

4. Eliminate negative exponents, then multiply numerator and denominator by $ 3(x^{2}-9){2/3} $ , the only internal denominator:

\[ \begin{aligned}\frac{(x^{2}-9)^{1/3}\ 3-(4x)\binom{\frac{1}{3}}{3}(x^{2}-9)^{-2/3}(2x)}{[(x^{2}-9)^{1/3}]^{2}}&=\frac{(x^{2}-9)^{1/3}\ 3-\frac{(4x)(2x)}{3(x^{2}-9)^{2/3}}}{(x^{2}-9)^{2/3}}\\&=\frac{(x^{2}-9)^{1/3}\ 3-\frac{(4x)(2x)}{3(x^{2}-9)^{2/3}}}{(x^{2}-9)^{2/3}}\cdot\frac{3(x^{2}-9)^{2/3}}{3(x^{2}-9)^{2/3}}\\&=\frac{9(x^{2}-9)-8x^{2}}{3(x^{2}-9)^{4/3}}\\&=\frac{x^{2}-81}{3(x^{2}-9)^{4/3}}\end{aligned} \]

4.12. Let \(z = 4 - 7i\) and \(w = -6 + 5i\) be two complex numbers. Find

1. \(z + w\) (b) \(w - z\) (c) \(wz\) (d) \(\frac{w}{z}\) (e) \(w^{2} - i\bar{z}\)

\[ \begin{aligned}(a)\quad z+w=(4-7i)+(-6+5i)=4-7i-6+5i=-2-2i\end{aligned} \]

\[ Use FOIL:wz=(-6+5i)(4-7i)=-24+42i+20i-35i^{2}=-24+62i+35=11+62i \]

4. To write the quotient of two complex numbers in standard form, multiply numerator and denominator of the quotient by the conjugate of the denominator:

\[ \frac{w}{z}=\frac{-6+5i}{4-7i}=\frac{-6+5i}{4-7i}\cdot\frac{4+7i}{4+7i}=\frac{-59-22i}{16-49i^{2}}=\frac{-59-22i}{16+49}=\frac{-59-22i}{65}\text{or}-\frac{59}{65}-\frac{22}{65}i \]

\[ \begin{aligned}(e)\quad&w^{2}-\bar{i}\bar{z}=(-6+5i)^{2}-\bar{i}\overline{(4-7i)}=36-60i+25i^{2}-i(4+7i)=36-60i-25-4i+7=18-64i\end{aligned} \]

SUPPLEMENTARY PROBLEMS

4.13. Reduce to lowest terms:

1. $ $ (b) $ $ (c) $ $ (d) $ $

Ans. (a) $ $ ; (b) $ $ ; (c) $ $ ; (d) $ 3x^{2}+3xh+h{2} $

4.14. Perform indicated operations:

\[ \begin{aligned}(a)\quad\frac{1}{(x+1)(x+2)}-\frac{3}{(x-1)(x+2)}+\frac{3}{(x-1)(x+1)}\quad(b)\frac{5}{x-2}+\frac{3}{x+2}-\frac{x-1}{x^{2}+4}\end{aligned} \]

\[ \begin{aligned}(c)\quad\frac{3x-1}{(x^{2}+4)^{2}}+\frac{2x-5}{x^{2}+4}\quad(d)(x^{2}-3x+2)\cdot\frac{x^{2}-5x+4}{x^{3}-6x^{2}+8x}\end{aligned} \]

\[ Ans.\quad(a)\ \frac{1}{(x-1)(x+1)};(b)\ \frac{7x^{3}+5x^{2}+36x+12}{x^{4}-16};(c)\ \frac{2x^{3}-5x^{2}+11x-21}{(x^{2}+4)^{2}};(d)\ \frac{x^{2}-2x+1}{x} \]

4.15. Write as a simple fraction in lowest terms:

\[ \begin{aligned}(a)\quad\frac{\frac{1}{t-1}+\frac{1}{t+1}}{\frac{1}{t}-\frac{1}{t^{2}}}\quad(b)\quad\frac{\frac{2x}{x+1}-\frac{2a}{a+1}}{x-a}\quad(c)\quad\frac{(x^{2}-4)^{3}(2x)-x^{2}(3)(x^{2}-4)^{2}(2x)}{(x^{2}-4)^{6}}\end{aligned} \]

\[ Ans.\quad(a)\ \frac{2t^{3}}{(t-1)(t^{2}-1)};(b)\ \frac{2}{(x+1)(a+1)};(c)\ \frac{-4x^{3}-8x}{(x^{2}-4)^{4}} \]

4.16. Write as a simple fraction in lowest terms:

\[ \begin{aligned}(a)\quad\frac{(x+5)^{-5}-(x+5)^{-4}}{(x+5)^{-3}};\quad(b)\frac{(x+h)^{-1}-x^{-1}}{h};\quad(c)\frac{x^{-2}-a^{-2}}{x-a}\end{aligned} \]

\[ Ans.\quad(a)\frac{-4-x}{(x+5)^{2}};(b)\frac{-1}{x(x+h)};(c)\frac{-x-a}{a^{2}x^{2}} \]

4.17. Write in simplest radical form: (a) $ $ (b) $ $ (c) $ $

Ans. (a) $ 2xyz^{{2}\sqrt[4]{3x}{2}y^{3}} $ ; (b) $ $ ; (c) $ $

4.18. Rationalize the denominator: (a) $ $ (b) $ $

Ans. (a) $ $ ; (b) $ $

4.19. Rationalize the numerator: (a) $ $ (b) $ $ (c) $ $

Ans. (a) $ $ ; (b) $ $ ; (c) $ $

4.20. Write as a sum or difference of terms in exponential notation: (a) $ $ ; (b) $ $

Ans. (a) $ 3x^{1/2}-2x{-1/2} $ ; (b) $ 2x^{8/3}-x{5/3}-4x^{2/3}+x{-1/3} $

4.21. Write as a simple fraction in lowest terms. Do not rationalize denominators.

\[ \frac{2x\sqrt{4-x^{2}}+\frac{x^{2}\cdot2x}{\sqrt{4-x^{2}}}}{4-x^{2}}\quad(b)\frac{\frac{2}{3}x(x^{2}+4)^{1/2}(x^{2}-9)^{-2/3}-x(x^{2}-9)^{1/3}(x^{2}+4)^{-1/2}}{x^{2}+4} \]

Ans. (a) $ $ ; (b) $ $

4.22. Write as a simple fraction in lowest terms. Do not rationalize denominators:

\[ \frac{x\left(\frac{1}{2}\right)\left(x^{2}+9\right)^{-1/2}\left(2x\right)-\sqrt{x^{2}+9}}{x^{2}}\quad(b)\frac{\left(x^{2}-1\right)^{3/2}-x\left(\frac{3}{2}\right)\left(x^{2}-1\right)^{1/2}\left(2x\right)}{\left(x^{2}-1\right)^{3}} \]

\[ \frac{(x^{2}-1)^{4/3}(2x)-(x^{2}+4)\left(\frac{4}{3}\right)(x^{2}-1)^{1/3}(2x)}{(x^{2}-1)^{8/3}} \]

\[ Ans.\quad(a)\ \frac{-9}{x^{2}(x^{2}+9)^{1/2}};(b)\ \frac{-1-2x^{2}}{(x^{2}-1)^{5/2}};(c)\ \frac{-2x^{3}-38x}{3(x^{2}-1)^{7/3}} \]

4.23. Let \(z = 5 - 2i\), \(w = -3 + i\). Write in standard form for complex numbers:

1. \(z + w\); (b) \(z - w\); (c) \(zw\); (d) \(z/w\)

Ans. (a) 2 - i; (b) 8 - 3i; (c) $ -13 + 11i $ ; (d) $ - + i $

4.24. Write in standard form for complex numbers:

1. $ $ (b) $ i^{6} $ (c) $ (1+2i)^{3} $ (d) $ (1-i)/(2+3i)-(4+5i)/(6i^{3}) $

Ans. (a) $ i $ ; (b) -1 or $ -1 + 0i $ ; (c) -11 - 2i; (d) $ - i $

4.25. For \(z\), \(w\) complex numbers, show:

1. \(\overline{z + w} = \overline{z} + \overline{w}\) (b) \(\overline{z - w} = \overline{z} - \overline{w}\) (c) \(\overline{zw} = \overline{z}\overline{w}\) (d) \(\overline{z/w} = \overline{z}/\overline{w}\)

2. \(\overline{z} = z\) if and only if \(z\) is a real number.