Rational Exponents, Radicals, and Complex Numbers Chapter 7 Rational Exponents, Radicals, and Complex Numbers

Radicals and Radical Functions § 7.1 Radicals and Radical Functions

Square Roots Opposite of squaring a number is taking the square root of a number. A number b is a square root of a number a if b2 = a. In order to find a square root of a, you need a # that, when squared, equals a.

Principal Square Roots Principal and Negative Square Roots If a is a nonnegative number, then is the principal or nonnegative square root of a is the negative square root of a.

Radicands Radical expression is an expression containing a radical sign. Radicand is the expression under a radical sign. Note that if the radicand of a square root is a negative number, the radical is NOT a real number.

Radicands Example:

Perfect Squares Square roots of perfect square radicands simplify to rational numbers (numbers that can be written as a quotient of integers). Square roots of numbers that are not perfect squares (like 7, 10, etc.) are irrational numbers. IF REQUESTED, you can find a decimal approximation for these irrational numbers. Otherwise, leave them in radical form.

Perfect Square Roots Radicands might also contain variables and powers of variables. To avoid negative radicands, assume for this chapter that if a variable appears in the radicand, it represents positive numbers only. Example:

Cube Roots Cube Root The cube root of a real number a is written as

Cube Roots Example:

nth Roots Other roots can be found, as well. The nth root of a is defined as If the index, n, is even, the root is NOT a real number when a is negative. If the index is odd, the root will be a real number.

nth Roots Example: Simplify the following.

nth Roots Example: Simplify the following. Assume that all variables represent positive numbers.

nth Roots If the index of the root is even, then the notation represents a positive number. But we may not know whether the variable a is a positive or negative value. Since the positive square root must indeed be positive, we might have to use absolute value signs to guarantee the answer is positive.

Finding nth Roots If n is an even positive integer, then If n is an odd positive integer, then

Finding nth Roots Simplify the following. If we know for sure that the variables represent positive numbers, we can write our result without the absolute value sign.

Finding nth Roots Example: Simplify the following. Since the index is odd, we don’t have to force the negative root to be a negative number. If a or b is negative (and thus changes the sign of the answer), that’s okay.

Evaluating Rational Functions We can also use function notation to represent rational functions. For example, Evaluating a rational function for a particular value involves replacing the value for the variable(s) involved. Example: Find the value

Root Functions Since every value of x that is substituted into the equation produces a unique value of y, the root relation actually represents a function. The domain of the root function when the index is even, is all nonnegative numbers. The domain of the root function when the index is odd, is the set of all real numbers.

Root Functions We have previously worked with graphing basic forms of functions so that you have some familiarity with their general shape. You should have a basic familiarity with root functions, as well.

Graphs of Root Functions Example: x y Graph (6, ) x y (4, 2) (2, ) (1, 1) 6 (0, 0) 4 2 2 1

Graphs of Root Functions Example: Graph x y x y 2 8 4 (8, 2) (4, ) (1, 1) 1 (-1, -1) (0, 0) (-4, ) -1 (-8, -2) -4 -2 -8

§ 7.2 Rational Exponents

Exponents with Rational Numbers So far, we have only worked with integer exponents. In this section, we extend exponents to rational numbers as a shorthand notation when using radicals. The same rules for working with exponents will still apply.

Understanding a1/n Recall that a cube root is defined so that However, if we let b = a1/3, then Since both values of b give us the same a, If n is a positive integer greater than 1 and is a real number, then

Using Radical Notation Example: Use radical notation to write the following. Simplify if possible.

Understanding am/n If m and n are positive integers greater than 1 with m/n in lowest terms, then as long as is a real number

Using Radical Notation Example: Use radical notation to write the following. Simplify if possible.

Understanding am/n as long as a-m/n is a nonzero real number.

Using Radical Notation Example: Use radical notation to write the following. Simplify if possible.

Using Rules for Exponents Example: Use properties of exponents to simplify the following. Write results with only positive exponents.

Using Rational Exponents Example: Use rational exponents to write as a single radical.

Simplifying Radical Expressions § 7.3 Simplifying Radical Expressions

Product Rule for Radicals If and are real numbers, then

Simplifying Radicals Example: Simplify the following radical expressions. No perfect square factor, so the radical is already simplified.

Simplifying Radicals Example: Simplify the following radical expressions.

Quotient Rule Radicals Quotient Rule for Radicals If and are real numbers,

Simplifying Radicals Example: Simplify the following radical expressions.

The Distance Formula Distance Formula The distance d between two points (x1,y1) and (x2,y2) is given by

The Distance Formula Example: Find the distance between (5, 8) and (2, 2).

The Midpoint Formula Midpoint Formula The midpoint of the line segment whose endpoints are (x1,y1) and (x2,y2) is the point with coordinates

The Midpoint Formula Example: Find the midpoint of the line segment that joins points P(5, 8) and P(2, 2).

Adding, Subtracting, and Multiplying Radical Expressions § 7.4 Adding, Subtracting, and Multiplying Radical Expressions

Sums and Differences Rules in the previous section allowed us to split radicals that had a radicand which was a product or a quotient. We can NOT split sums or differences.

Like Radicals In previous chapters, we’ve discussed the concept of “like” terms. These are terms with the same variables raised to the same powers. They can be combined through addition and subtraction. Similarly, we can work with the concept of “like” radicals to combine radicals with the same radicand. Like radicals are radicals with the same index and the same radicand. Like radicals can also be combined with addition or subtraction by using the distributive property.

Adding and Subtracting Radical Expressions Example: Can not simplify Can not simplify

Adding and Subtracting Radical Expressions Example: Simplify the following radical expression.

Adding and Subtracting Radical Expressions Example: Simplify the following radical expression.

Adding and Subtracting Radical Expressions Example: Simplify the following radical expression. Assume that variables represent positive real numbers.

Multiplying and Dividing Radical Expressions If and are real numbers,

Multiplying and Dividing Radical Expressions Example: Simplify the following radical expressions.

Rationalizing Numerators and Denominators of Radical Expressions § 7.5 Rationalizing Numerators and Denominators of Radical Expressions

Rationalizing the Denominator Many times it is helpful to rewrite a radical quotient with the radical confined to ONLY the numerator. If we rewrite the expression so that there is no radical in the denominator, it is called rationalizing the denominator. This process involves multiplying the quotient by a form of 1 that will eliminate the radical in the denominator.

Rationalizing the Denominator Example: Rationalize the denominator.

Conjugates Many rational quotients have a sum or difference of terms in a denominator, rather than a single radical. In that case, we need to multiply by the conjugate of the numerator or denominator (which ever one we are rationalizing). The conjugate uses the same terms, but the opposite operation (+ or ).

Rationalizing the Denominator Example: Rationalize the denominator.

Rationalizing the Numerator An expression rewritten with no radical in the numerator is called rationalizing the numerator. Example:

Rationalizing the Numerator Example: Rationalize the numerator.

Radical Equations and Problem Solving § 7.6 Radical Equations and Problem Solving

The Power Rule Power Rule If both sides of an equation are raised to the same power, solutions of the new equation contain all the solutions of the original equation, but might also contain additional solutions. A proposed solution of the new equation that is NOT a solution of the original equation is an extraneous solution.

Solving Radical Equations Solving a Radical Equation Isolate one radical on one side of the equation. Raise each side of the equation to a power equal to the index of the radical and simplify. If the equation still contains a radical term, repeat Steps 1 and 2. If not, solve the equation. Check all proposed solutions in the original equation.

Solving Radical Equations Example: Solve the following radical equation. Substitute into the original equation. true So the solution is x = 2.

Solving Radical Equations Example: Solve the following radical equation.

Solving Radical Equations Example continued: Substitute the value for x into the original equation, to check the solution. true So the solution is x = 3. false

Solving Radical Equations Example: Solve the following radical equation.

Solving Radical Equations Example continued: Substitute the value for x into the original equation, to check the solution. false So the solution is .

Solving Radical Equations Example: Solve the following radical equation.

Solving Radical Equations Example continued: Substitute the value for x into the original equation, to check the solution. true true So the solution is x = 4 or 20.

Solving Radical Equations Example: Solve the following radical equation. Substitute into the original equation. true So the solution is x = 24.

Solving Radical Equations Example: Solve the following radical equation. Substitute into the original equation. Does NOT check, since the left side of the equation is asking for the principal square root. So the solution is .

The Pythagorean Theorem If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then a2 + b2 = c2 There are several applications in this section that require the use of the Pythagorean Theorem in order to solve.

Using the Pythagorean Theorem Example: Find the length of the hypotenuse of a right triangle when the length of the two legs are 2 inches and 7 inches. c2 = 22 + 72 = 4 + 49 = 53 c = inches

§ 7.7 Complex Numbers

Imaginary Numbers Previously, when we encountered square roots of negative numbers in solving equations, we would say “no real solution” or “not a real number”. Imaginary Unit The imaginary unit i, is the number whose square is – 1. That is,

The Imaginary Unit, i Example: Write the following with the i notation. i i i i

Complex Numbers Real numbers and imaginary numbers are both subsets of a new set of numbers. Complex Numbers A complex number is a number that can be written in the form a + bi, where a and b are real numbers.

Standard Form of Complex Numbers Complex numbers can be written in the form a + bi (called standard form), with both a and b as real numbers. a is a real number and bi would be an imaginary number. If b = 0, a + bi is a real number. If a = 0, a + bi is an imaginary number.

Standard Form of Complex Numbers Example: Write each of the following in the form of a complex number in standard form a + bi. 6 = 6 + 0i 8i = 0 + 8i i i 6 + 5i

Adding and Subtracting Complex Numbers Sum or Difference of Complex Numbers If a + bi and c + di are complex numbers, then their sum is (a + bi) + (c + di) = (a + c) + (b + d)i Their difference is (a + bi) – (c + di) = (a – c) + (b – d)i

Adding and Subtracting Complex Numbers Example: Add or subtract the following complex numbers. Write the answer in standard form a + bi. (4 + 6i) + (3 – 2i) = (4 + 3) + (6 – 2)i = 7 + 4i (8 + 2i) – (4i) = (8 – 0) + (2 – 4)i = 8 – 2i

Multiplying Complex Numbers The technique for multiplying complex numbers varies depending on whether the numbers are written as single term (either the real or imaginary component is missing) or two terms.

Multiplying Complex Numbers Note that the product rule for radicals does NOT apply for imaginary numbers.

Multiplying Complex Numbers Example: Multiply the following complex numbers. 8i · 7i 56i2 56(1) 56

Multiplying Complex Numbers Example: Multiply the following complex numbers. Write the answer in standard form a + bi. 5i(4 – 7i) 20i – 35i2 20i – 35(–1) 20i + 35 35 + 20i

Multiplying Complex Numbers Example: Multiply the following complex numbers. Write the answer in standard form a + bi. (6 – 3i)(7 + 4i) 42 + 24i – 21i – 12i2 42 + 3i – 12(–1) 42 + 3i + 12 54 + 3i

Complex Conjugate Complex Conjugates The complex numbers (a + bi) and (a – bi) are complex conjugates of each other, and (a + bi)(a – bi) = a2 + b2

Complex Conjugate The conjugate of a + bi is a – bi. The product of (a + bi) and (a – bi) is (a + bi)(a – bi) a2 – abi + abi – b2i2 a2 – b2(–1) a2 + b2, which is a real number.

Dividing Complex Numbers Example: Use complex conjugates to divide the following complex numbers. Write the answer in standard form.

Dividing Complex Numbers Example: Divide the following complex numbers.

Patterns of i The powers recycle through each multiple of 4.

Patterns of i Example: Simplify each of the following powers.