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Учебные пособия > Boundless Algebra

Radicals

Introduction to Radicals

Radical expressions yield roots and are the inverse of exponential expressions.

Learning Objectives

Describe the root of a number in terms of exponentiation

Key Takeaways

Key Points

  • Roots are the inverse operation of exponentiation. This means that if [latex]\sqrt [ n ]{ x } = r[/latex], then [latex]{r}^{n}=x[/latex].
  • The square root of a value is the number that when squared results in the initial value. In other words, [latex]\sqrt{y}=x[/latex] if [latex]x^2=y[/latex].
  • The cube root of a value is the number that when cubed results in the initial value. In other words, [latex]\sqrt[3]{y} = x[/latex] if [latex]x^3 = y[/latex].

Key Terms

  • root: A number that when raised to a specified power yields a specified number or expression.
  • radical expression: A mathematical expression that contains a root, written in the form [latex]\sqrt[n]{a}[/latex].
  • cube root: A root of degree 3, written in the form [latex]\sqrt[3]{a}[/latex].
  • square root: A root of degree 2, written in the form [latex]\sqrt{a}[/latex].
Roots are the inverse operation of exponentiation. Mathematical expressions with roots are called radical expressions and can be easily recognized because they contain a radical symbol ([latex]\sqrt{}[/latex]). Recall that exponents signify that we should multiply a given integer a certain number of times. For example, [latex]7^2[/latex] tells us that we should multiply 7 by itself two times: [latex-display]7^2 = 7 \cdot 7 = 49[/latex-display] Since roots are the inverse operation of exponentiation, they allow us to work backwards from the solution of an exponential expression to the number in the base of the expression. For example, the following is a radical expression that reverses the above solution, working backwards from 49 to its square root: [latex-display]\sqrt{49} = 7[/latex-display] In this expression, the symbol is known as the "radical," and the solution of 7 is called the "root." Finding the value for a particular root can be much more difficult than solving an exponential expression. For now, it is important simplify to recognize the relationship between roots and exponents: if a root [latex]r[/latex] is defined as the [latex]n \text{th}[/latex] root of [latex]x[/latex], it is represented as [latex-display]\sqrt [ n ]{ x } = r[/latex-display] Because roots are the inverse of exponents, we can cancel out the root in this equation by raising the answer to the nth power: [latex-display]\left( \sqrt [ n ]{ x }\right) ^n = \left(r\right) ^n[/latex-display] To simplify: [latex-display]{r}^{n}=x[/latex-display]

Square Roots

If the square root of a number [latex]x[/latex] is calculated, the result is a number that when squared (i.e., when raised to an exponent of 2) gives the original number [latex]x[/latex]. This can be written symbolically as follows: [latex]\sqrt x = y[/latex] if [latex]{y}^{2}=x[/latex]. This rule applies to the series of real numbers [latex]{ y }^{ 2 }\ge 0[/latex], regardless of the value of [latex]y[/latex]. As such, when [latex]x<0[/latex] then [latex]\sqrt x[/latex] cannot be defined. For example, consider the following: [latex]\sqrt{36}[/latex]. This is read as "the square root of 36" or "radical 36." You may recognize that [latex]6^2 = 6 \cdot 6 = 36[/latex], and therefore conclude that 6 is the root of [latex]\sqrt{36}[/latex]. Thus we have the answer, [latex]\sqrt{36} = 6[/latex].

Cube Roots

The cube root of a number ([latex]\sqrt [ 3 ]{x}[/latex]) can also be calculated. The cube root of a value [latex]x[/latex] is the number that when cubed (i.e., when raised to an exponent of 3) yields the original number [latex]x[/latex]. For example, the cube root of 8 is 2 because [latex]2^3 = 2 \cdot 2\cdot 2=8[/latex]. This can also be written as [latex]\sqrt[3]{8}=2[/latex].

Other Roots

There are an infinite number of possible roots all in the form of [latex]\sqrt [n]{a}[/latex]. Any non-zero integer can be substituted for [latex]n[/latex]. For example, [latex]\sqrt[4]{a}[/latex] is called the "fourth root of [latex]a[/latex]," and [latex]\sqrt[20]{a}[/latex] is called the "twentieth root of [latex]a[/latex]." Note that for any such root, if [latex]\sqrt [n]{a} = b[/latex] then [latex]{b}^{n} = a[/latex]. As an example, consider [latex]\sqrt[4]{2401} = 7[/latex]. [latex]7^4 = 7\cdot 7\cdot 7\cdot 7 = 2401[/latex].

Adding, Subtracting, and Multiplying Radical Expressions

Radicals and exponents have particular requirements for addition and subtraction while multiplication is carried out more freely.

Learning Objectives

Differentiate between correct and incorrect uses of operations on radical expressions

Key Takeaways

Key Points

  • To add radicals, the radicand (the number that is under the radical) must be the same for each radical.
  • Subtraction follows the same rules as addition: the radicand must be the same.
  • Multiplication of radicals simply requires that we multiply the term under the radical signs.

Key Terms

  • radicand: The number or expression whose square root or other root is being considered; e.g., the 3 in [latex]\sqrt[n]{3}[/latex]. More simply, the number under the radical.
  • radical expression: An expression that represents the root of a number or quantity.
Roots are the inverse operation for exponents. An expression with roots is called a radical expression. It's easy, although perhaps tedious, to compute exponents given a root. For instance [latex]7\cdot7\cdot7\cdot7 = 49\cdot49 = 2401[/latex]. So, we know the fourth root of 2401 is 7, and the square root of 2401 is 49. What is the third root of 2401? Finding the value for a particular root is difficult. This is because exponentiation is a different kind of function than addition, subtraction, multiplication, and division. Let's go through some basic mathematical operations with radicals and exponents.

Adding and Subtracting Radical Expressions

To add radicals, the radicand (the number that is under the radical) must be the same for each radical, so, a generic equation will have the form: [latex-display]a\sqrt{b}+c\sqrt{b} = (a+c)\sqrt{b}[/latex-display] Let's plug some numbers in place of the variables: [latex-display]\sqrt 3 +2\sqrt 3 = 3\sqrt 3[/latex-display] Subtraction follows the same rules as addition: [latex-display]a\sqrt b - c\sqrt b = (a-c)\sqrt b[/latex-display] For example: [latex-display]3\sqrt 3 -2\sqrt 3 = \sqrt 3[/latex-display]

Multiplying Radical Expressions

Multiplication of radicals simply requires that we multiply the variable under the radical signs. [latex-display]\sqrt a \cdot \sqrt b = \sqrt {a\cdot b}[/latex-display] Some examples with real numbers: [latex-display]\sqrt 3 \cdot \sqrt 6 = \sqrt {18}[/latex-display] This equation can actually be simplified further; we will go over simplification in another section.

Simplifying Radical Expressions

A radical expression can be simplified if:
  1. the value under the radical sign can be written as an exponent,
  2. there are fractions under the radical sign,
  3. there is a radical expression in the denominator.
For example, the radical expression [latex]\displaystyle \sqrt{\frac{16}{3}}[/latex] can be simplified by first removing the squared value from the numerator. [latex-display]\displaystyle \sqrt{\frac{16}{3}} = \sqrt{\frac{4^2}{3}} = 4\sqrt{\frac{1}{3}}[/latex-display] Then, the fraction under the radical sign can be addressed, and the radical in the numerator can again be simplified. [latex-display]\displaystyle 4\sqrt{\frac{1}{3}} = \frac{4\sqrt{1}}{\sqrt{3}} = \frac{4}{\sqrt{3}}[/latex-display] Finally, the radical needs to be removed from the denominator. [latex-display]\displaystyle \frac{4}{\sqrt{3}} = \frac{4}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{4\sqrt{3}}{3} = \frac{4}{3}\sqrt{3}[/latex-display]

Fractions Involving Radicals

Root rationalization is a process by which any roots in the denominator of an irrational fraction are eliminated.

Learning Objectives

Convert between fractions with and without rationalized denominators

Key Takeaways

Key Points

  • To rationalize the denominator, multiply both the numerator and denominator by the radical in the denominator.

Key Terms

  • rationalization: A process by which radicals in the denominator of an fraction are eliminated.
In mathematics, we are often given terms in the form of fractions with radicals in the numerator and/or denominator. When we are given expressions that involve radicals in the denominator, it makes it easier to evaluate the expression if we rewrite it in a way that the radical is no longer in the denominator. This process is called rationalizing the denominator. Before we begin, remember that whatever we do to one side of an algebraic equation, we must also do to the other side. This same principle can be applied to fractions: whatever we do to the numerator, we must also do to the denominator, and vice versa. Let's look at an example to illustrate the process of rationalizing the denominator. You are given the fraction [latex]\frac{10}{\sqrt{3}}[/latex], and you want to simplify it by eliminating the radical from the denominator. Recall that a radical multiplied by itself equals its radicand, or the value under the radical sign. Therefore, multiply the top and bottom of the fraction by [latex]\frac{\sqrt{3}}{\sqrt{3}}[/latex], and watch how the radical expression disappears from the denominator:[latex]\displaystyle \frac{10}{\sqrt{3}} \cdot\frac{\sqrt{3}}{\sqrt{3}} = {\frac{10\cdot\sqrt{3}}{{\sqrt{3}}^2}} = {\frac{10\sqrt{3}}{3}}[/latex]

Imaginary Numbers

There is no such value such that when squared it results in a negative value; we therefore classify roots of negative numbers as "imaginary."

Learning Objectives

Explain what imaginary numbers are and why they are needed in mathematics

Key Takeaways

Key Points

  • There is no such value such that when squared it results in a negative value.More specifically, solving [latex]x^2=-1[/latex] for [latex]x[/latex] results in a "number" that would not be a real number, referred to as an imaginary number.
  • The imaginary number, [latex]i[/latex], is defined as the square root of -1: [latex]i=\sqrt{-1}[/latex].

Key Terms

  • imaginary number: The square root of -1.
  • radicand: The value under the radical sign.
A radical expression represents the root of a given quantity. What does it mean, then, if the value under the radical is negative, such as in [latex]\displaystyle \sqrt{-1}[/latex]? There is no real value such that when multiplied by itself it results in a negative value. This means that there is no real value of [latex]x[/latex] that would make [latex]x^2 =-1[/latex] a true statement. That is where imaginary numbers come in. When the radicand (the value under the radical sign) is negative, the root of that value is said to be an imaginary number. Specifically, the imaginary number, [latex]i[/latex], is defined as the square root of -1: thus, [latex]i=\sqrt{-1}[/latex]. We can write the square root of any negative number in terms of [latex]i[/latex]. Here are some examples:
  • [latex]\sqrt{-25}=\sqrt{25\cdot-1}=\sqrt{25}\cdot\sqrt{-1}=5i[/latex]
  • [latex]\sqrt{-18} = \sqrt{2\cdot9\cdot-1} = \sqrt{2} \cdot \sqrt{9} \cdot \sqrt{-1} = 3i\sqrt{2}[/latex]

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