Glossary

1.2 Exponents and Radicals

4 min read•july 31, 2024

Exponents and radicals are powerful tools in algebra, letting us express and manipulate numbers in compact ways. They're key to understanding growth, decay, and roots in math and science. Mastering these concepts opens doors to more advanced math.

From basic exponent rules to solving complex radical equations, this topic builds a strong foundation for algebra. It teaches us to simplify expressions, perform operations, and solve equations with these important mathematical concepts.

Simplifying expressions with exponents

Laws of exponents

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  • The product rule adds exponents when multiplying exponential expressions with the same base: am∗an=a(m+n)a^m * a^n = a^(m+n)
    • Example: 23∗24=2(3+4)=27=1282^3 * 2^4 = 2^(3+4) = 2^7 = 128
  • The quotient rule subtracts exponents when dividing exponential expressions with the same base: am/an=a(m−n)a^m / a^n = a^(m-n)
    • Example: 35/32=3(5−2)=33=273^5 / 3^2 = 3^(5-2) = 3^3 = 27
  • The power rule multiplies exponents when an exponential expression is raised to a power: (am)n=a(m∗n)(a^m)^n = a^(m*n)
    • Example: (42)3=4(2∗3)=46=4,096(4^2)^3 = 4^(2*3) = 4^6 = 4,096

Special cases of exponents

  • A number raised to the power of zero equals one: a0=1a^0 = 1, where a≠0a ≠ 0
    • Example: 50=15^0 = 1
  • A number raised to a equals its reciprocal raised to the : a(−n)=1/ana^(-n) = 1 / a^n
    • Example: 2(−3)=1/23=1/82^(-3) = 1 / 2^3 = 1/8
  • When a product is raised to a power, each factor is raised to that power: (ab)n=an∗bn(ab)^n = a^n * b^n
    • Example: (2∗3)4=24∗34=16∗81=1,296(2 * 3)^4 = 2^4 * 3^4 = 16 * 81 = 1,296
  • When a quotient is raised to a power, the numerator and denominator are each raised to that power: (a/b)n=an/bn(a/b)^n = a^n / b^n, where b≠0b ≠ 0
    • Example: (6/2)2=62/22=36/4=9(6/2)^2 = 6^2 / 2^2 = 36 / 4 = 9

Rational exponents and radicals

Converting between rational exponents and radicals

  • The nth root of a number aa can be written as a radical expression: an\sqrt[n]{a}, or as an exponential expression with a : a(1/n)a^(1/n)
  • When converting from a radical to a rational exponent, the index of the radical becomes the denominator of the exponent: an=a(1/n)\sqrt[n]{a} = a^(1/n)
    • Example: 83=8(1/3)=2\sqrt[3]{8} = 8^(1/3) = 2
  • When converting from a rational exponent to a radical, the denominator of the exponent becomes the index of the radical: a(m/n)=amna^(m/n) = \sqrt[n]{a^m}
    • Example: 16(3/4)=1634=4,0964=816^(3/4) = \sqrt[4]{16^3} = \sqrt[4]{4,096} = 8

Expressions with coefficients and radicals

  • If an expression has both a coefficient and a radical, the coefficient should be raised to the power of the rational exponent: b∗an=(bn∗a)(1/n)b * \sqrt[n]{a} = (b^n * a)^(1/n)
    • Example: 2∗273=(23∗27)(1/3)=216(1/3)=62 * \sqrt[3]{27} = (2^3 * 27)^(1/3) = 216^(1/3) = 6

Operations with radicals and exponents

Multiplying and dividing radicals

  • When multiplying expressions with the same root index, multiply the radicands and simplify: an∗bn=abn\sqrt[n]{a} * \sqrt[n]{b} = \sqrt[n]{ab}
    • Example: 3∗12=36=6\sqrt{3} * \sqrt{12} = \sqrt{36} = 6
  • When dividing expressions with the same root index, divide the radicands and simplify: an/bn=a/bn\sqrt[n]{a} / \sqrt[n]{b} = \sqrt[n]{a/b}, where b≠0b ≠ 0
    • Example: 18/2=18/2=9=3\sqrt{18} / \sqrt{2} = \sqrt{18/2} = \sqrt{9} = 3

Adding and subtracting radicals

  • To add or subtract radical expressions, they must have the same root index and radicand. If they do, add or subtract the coefficients: a∗cn±b∗cn=(a±b)∗cna * \sqrt[n]{c} ± b * \sqrt[n]{c} = (a ± b) * \sqrt[n]{c}
    • Example: 25+35=(2+3)5=552\sqrt{5} + 3\sqrt{5} = (2 + 3)\sqrt{5} = 5\sqrt{5}
  • When simplifying expressions with rational exponents, use the laws of exponents to perform arithmetic operations and simplify the result
    • Example: 2(1/3)∗2(2/3)=2((1/3)+(2/3))=21=22^(1/3) * 2^(2/3) = 2^((1/3) + (2/3)) = 2^1 = 2

Solving radical and exponential equations

Solving equations with a single radical

  • To solve an equation containing a single radical, isolate the radical on one side of the equation and raise both sides to the power of the root index
    • Example: x+1=3\sqrt{x + 1} = 3
      • (x+1)2=32(\sqrt{x + 1})^2 = 3^2
      • x+1=9x + 1 = 9
      • x=8x = 8

Solving equations with multiple radicals

  • When solving an equation with multiple radicals, isolate one radical at a time and solve for the variable. Check the solution by substituting it back into the original equation
    • Example: x+x−2=3\sqrt{x} + \sqrt{x - 2} = 3
      • x=3−x−2\sqrt{x} = 3 - \sqrt{x - 2}
      • (x)2=(3−x−2)2(\sqrt{x})^2 = (3 - \sqrt{x - 2})^2
      • x=9−6x−2+x−2x = 9 - 6\sqrt{x - 2} + x - 2
      • 0=7−6x−20 = 7 - 6\sqrt{x - 2}
      • 6x−2=76\sqrt{x - 2} = 7
      • x−2=7/6\sqrt{x - 2} = 7/6
      • (x−2)2=(7/6)2(\sqrt{x - 2})^2 = (7/6)^2
      • x−2=49/36x - 2 = 49/36
      • x=85/36x = 85/36

Solving equations with rational exponents

  • To solve an equation containing rational exponents, first isolate the term with the variable and rational exponent on one side of the equation
  • Next, raise both sides of the equation to the reciprocal of the rational exponent to eliminate the exponent and solve for the variable
  • Check the solution by substituting it back into the original equation. Reject any extraneous solutions that do not satisfy the original equation
    • Example: 2x(2/3)−5=32x^(2/3) - 5 = 3
      • 2x(2/3)=82x^(2/3) = 8
      • x(2/3)=4x^(2/3) = 4
      • (x(2/3))(3/2)=4(3/2)(x^(2/3))^(3/2) = 4^(3/2)
      • x=8x = 8

Key Terms to Review (16)

Adding exponents: Adding exponents refers to the process of combining terms with the same base and different exponents through addition. This concept is essential for simplifying expressions in algebra, especially when working with exponential functions. It helps in understanding how to manipulate powers and express them in a more manageable form, especially when dealing with polynomials or simplifying radical expressions.
Cube root: The cube root of a number is a value that, when multiplied by itself three times, gives the original number. This concept is closely tied to exponents and radicals, where the cube root can be expressed using fractional exponents as $$x^{1/3}$$ and represented in radical form as $$\sqrt[3]{x}$$. Understanding cube roots is essential for solving equations involving cubes and is foundational in the study of polynomial functions.
Equation of Exponential Form: An equation of exponential form expresses a relationship where a constant base is raised to a variable exponent, typically written as $$a^x = b$$, where 'a' is the base, 'x' is the exponent, and 'b' is the result. This form is crucial for solving problems involving growth or decay, particularly in contexts like finance or biology. Understanding this equation allows for converting between different forms and helps in graphing exponential functions effectively.
Exponential Function: An exponential function is a mathematical expression in the form of $$f(x) = a imes b^x$$, where 'a' is a constant, 'b' is the base that is a positive real number not equal to 1, and 'x' is the exponent. This type of function demonstrates rapid growth or decay as the value of 'x' changes, making it crucial for modeling various real-world phenomena such as population growth and radioactive decay. The behavior of exponential functions connects closely with logarithmic functions, providing insight into how these two concepts interact in mathematics and applications.
Exponential Growth: Exponential growth refers to a process where the quantity increases at a rate proportional to its current value, leading to rapid increases over time. This phenomenon is commonly represented by an exponential function of the form $$f(t) = a imes b^{t}$$, where 'a' is the initial amount, 'b' is the growth factor, and 't' is time. Understanding exponential growth is crucial as it appears in various contexts, such as population dynamics, finance, and natural phenomena.
Index of a radical: The index of a radical indicates the root being taken when simplifying or evaluating a radical expression. It is typically represented as a small number positioned to the upper left of the radical sign, showing how many times a number must be multiplied by itself to achieve the value under the radical. Understanding the index helps in determining both the nature of the roots and how they interact with exponents.
Logarithmic scale: A logarithmic scale is a type of scale used to represent large ranges of values by using the logarithm of the value instead of the value itself. This scale helps to compress wide-ranging data into a more manageable format, making it easier to visualize and interpret exponential relationships, such as those seen in exponential growth or decay.
Negative exponent: A negative exponent indicates that a number should be taken as a reciprocal and raised to the corresponding positive exponent. This means that if you have a term like $a^{-n}$, it can be rewritten as $\frac{1}{a^{n}}$. Understanding negative exponents is crucial for simplifying expressions, especially when combined with other exponent rules such as multiplication and division.
Positive Exponent: A positive exponent indicates how many times a base is multiplied by itself. For example, in the expression $$a^n$$, where $$n$$ is a positive integer, the base $$a$$ is multiplied by itself $$n$$ times. Understanding positive exponents is crucial for manipulating expressions involving powers and is foundational for more complex concepts involving exponents and radicals.
Power of a power: The power of a power refers to an exponentiation rule that states when raising a power to another power, you multiply the exponents. For example, if you have $(a^m)^n$, the result can be simplified to $a^{m \cdot n}$. This rule is vital for simplifying expressions and solving equations involving exponents, making it a fundamental concept in understanding exponent rules.
Product of Powers: The product of powers is a property of exponents that states when multiplying two expressions with the same base, you can add the exponents. This can be expressed as $$a^m \cdot a^n = a^{m+n}$$, where 'a' is the base and 'm' and 'n' are the exponents. Understanding this concept is crucial when simplifying expressions and solving equations that involve exponentiation, as it allows for efficient manipulation of terms with the same base.
Radical Equation: A radical equation is an equation in which at least one variable is under a radical sign, such as a square root, cube root, or any other root. These types of equations can often be solved by isolating the radical and then raising both sides of the equation to the power that eliminates the radical. Understanding radical equations involves recognizing the properties of exponents and radicals, as well as applying techniques for solving them correctly.
Rational Exponent: A rational exponent is an exponent that can be expressed as a fraction, where the numerator represents the power and the denominator indicates the root. This concept links the operation of exponentiation with that of taking roots, allowing for expressions to be simplified and manipulated more easily. Rational exponents provide a unified approach to expressing both powers and roots, making it easier to work with various mathematical expressions.
Scientific Notation: Scientific notation is a way of expressing very large or very small numbers in a compact form, typically in the format of $a \times 10^n$, where 'a' is a number greater than or equal to 1 and less than 10, and 'n' is an integer. This method simplifies calculations and comparisons by transforming cumbersome numbers into manageable formats. It connects closely with concepts of exponents, where the exponent indicates the power of ten by which the coefficient is multiplied.
Simplifying radicals: Simplifying radicals involves the process of reducing a radical expression to its simplest form, where no perfect square factors remain under the radical sign. This process connects deeply with the concepts of exponents, as radicals can be expressed as fractional exponents. Recognizing relationships between numbers, like perfect squares and cubes, helps in simplifying these expressions effectively.
Square Root: A square root is a value that, when multiplied by itself, gives the original number. In mathematical notation, the square root of a number 'x' is represented as $$\sqrt{x}$$. Square roots are closely related to exponents, as the square root of a number can be expressed as raising that number to the power of one-half, or $$x^{1/2}$$. This connection helps to understand how square roots operate within the broader context of exponents and radicals.
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