Glossary

1.3 Radicals and Rational Exponents

3 min read•june 24, 2024

Radicals and rational exponents are key tools for simplifying complex expressions in algebra. They allow us to work with numbers that aren't or cubes, opening up new ways to solve equations and model real-world scenarios.

Understanding these concepts is crucial for tackling more advanced math topics. From simplifying square roots to manipulating rational exponents, these skills form the foundation for working with , , and beyond.

Radicals

Simplification of square roots

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  • Definition: a\sqrt{a} is the number which when multiplied by itself equals aa
  • Simplify by factoring out perfect squares ab=a⋅b\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}
    • Perfect squares simplify to the base a2=a\sqrt{a^2} = a for a≥0a \geq 0 (4, 9, 16, 25)
  • Estimate square roots by identifying the nearest perfect squares less than and greater than the
    • Estimate the value between these two perfect squares (10 is between 9 and 16)
  • The result of simplifying a may be an irrational number

Product and quotient rules for radicals

  • multiplies radicals with the same a⋅b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}
    • Simplify radicals before applying the product rule (8⋅18=22⋅32=62\sqrt{8} \cdot \sqrt{18} = 2\sqrt{2} \cdot 3\sqrt{2} = 6\sqrt{2})
  • divides radicals with the same ab=ab\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} for b≠0b \neq 0
    • Simplify radicals before applying the quotient rule (502=522=5\frac{\sqrt{50}}{\sqrt{2}} = \frac{5\sqrt{2}}{\sqrt{2}} = 5)
  • Simplify complex using both product and quotient rules (123⋅8=233⋅22=42\frac{\sqrt{12}}{\sqrt{3}} \cdot \sqrt{8} = \frac{2\sqrt{3}}{\sqrt{3}} \cdot 2\sqrt{2} = 4\sqrt{2})

Addition and subtraction of square roots

  • have the same , add or subtract coefficients ab±cb=(a±c)ba\sqrt{b} \pm c\sqrt{b} = (a \pm c)\sqrt{b}
    • 25+35=552\sqrt{5} + 3\sqrt{5} = 5\sqrt{5}
  • have different radicands, simplify each separately
    • Express the result as the sum or difference of the simplified radicals (8−18=22−32=−2\sqrt{8} - \sqrt{18} = 2\sqrt{2} - 3\sqrt{2} = -\sqrt{2})

Rationalization of radical denominators

  • Rationalizing a single term denominator: multiply numerator and denominator by the denominator's conjugate
    • 12=12⋅22=22\frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}
  • Rationalizing a two term denominator (): multiply numerator and denominator by the conjugate of the denominator
    • 13+2=13+2⋅3−23−2=3−23−2=3−2\frac{1}{\sqrt{3} + \sqrt{2}} = \frac{1}{\sqrt{3} + \sqrt{2}} \cdot \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} = \frac{\sqrt{3} - \sqrt{2}}{3 - 2} = \sqrt{3} - \sqrt{2}

Rational Exponents

Simplification of rational exponents

  • Definition: amn=amna^{\frac{m}{n}} = \sqrt[n]{a^m} for a>0a > 0 if nn is even
  • Properties:
    1. Product rule amn⋅apq=amq+npnqa^{\frac{m}{n}} \cdot a^{\frac{p}{q}} = a^{\frac{mq + np}{nq}}
    2. Quotient rule amnapq=amq−npnq\frac{a^{\frac{m}{n}}}{a^{\frac{p}{q}}} = a^{\frac{mq - np}{nq}}
    3. (amn)pq=ampnq(a^{\frac{m}{n}})^{\frac{p}{q}} = a^{\frac{mp}{nq}}
  • Simplify expressions containing rational exponents using these properties (412⋅413=41⋅3+2⋅12⋅3=456=2464^{\frac{1}{2}} \cdot 4^{\frac{1}{3}} = 4^{\frac{1 \cdot 3 + 2 \cdot 1}{2 \cdot 3}} = 4^{\frac{5}{6}} = 2\sqrt[6]{4})

Conversion between radicals and exponents

  • Converting from to form:
    • an=a1n\sqrt[n]{a} = a^{\frac{1}{n}}
    • amn=amn\sqrt[n]{a^m} = a^{\frac{m}{n}}
  • Converting from rational exponent to radical form:
    • a1n=ana^{\frac{1}{n}} = \sqrt[n]{a}
    • amn=amna^{\frac{m}{n}} = \sqrt[n]{a^m}
  • Simplify expressions by converting between radical and rational exponent forms (814=8114=(34)14=3\sqrt[4]{81} = 81^{\frac{1}{4}} = (3^4)^{\frac{1}{4}} = 3)

Advanced Concepts

Complex Numbers and Absolute Value

  • Complex numbers involve the square root of negative numbers (e.g., −1=i\sqrt{-1} = i)
  • The of a complex number is its distance from zero on the complex plane

Domain Restrictions and Exponential Functions

  • Radicals and rational exponents often have to ensure real-valued results
  • Exponential functions, such as f(x)=axf(x) = a^x, are related to rational exponents and have specific properties and applications

Key Terms to Review (40)

Absolute value: Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is always non-negative and is denoted by vertical bars, e.g., $|x|$.
Absolute Value: Absolute value is a mathematical concept that represents the distance of a number from zero on the number line, regardless of the number's sign. It is a fundamental operation that is essential in understanding and working with real numbers, radicals, and sequences.
Complex Numbers: Complex numbers are a mathematical concept that extend the real number system by including the imaginary unit, denoted as $i$, which is defined as the square root of -1. They are used to represent quantities that have both magnitude and direction, and are essential in various areas of mathematics, including algebra, calculus, and physics.
Conjugate Radical: A conjugate radical is a pair of radicals that differ only in the sign of one of their unpaired electrons. These radicals are formed when a single covalent bond is broken, resulting in two radicals with opposite spins on the shared atom.
Cube Root: The cube root is a mathematical operation that finds the value which, when multiplied by itself three times, equals a given number. It is a type of radical operation that is closely related to the concepts of radicals and rational exponents.
Domain Restrictions: Domain restrictions refer to the limitations or constraints placed on the set of input values or independent variables that a function or equation can accept. These restrictions define the valid range of values for which the function or equation is defined and can be meaningfully evaluated.
Exponent Rule: An exponent rule is a mathematical principle that describes how to perform operations involving exponents. Exponents are used to represent repeated multiplication of a number, and exponent rules provide a systematic way to manipulate and simplify expressions containing exponents.
Exponential Functions: An exponential function is a mathematical function in which the independent variable appears as an exponent. These functions model situations where a quantity grows or decays at a constant rate over time, and they are characterized by an initial value and a constant growth or decay factor.
Index: In the context of radicals, the index specifies the root being taken. For example, in $\sqrt[n]{x}$, 'n' is the index and indicates an n-th root.
Index: An index is a numerical or alphabetical representation that indicates the position or location of a specific value or element within a set, such as a list, array, or mathematical expression. It serves as a reference point to access or identify a particular item or piece of information.
Irrational numbers: Irrational numbers are real numbers that cannot be expressed as the ratio of two integers. Their decimal expansions are non-repeating and non-terminating.
Irrational Numbers: Irrational numbers are real numbers that cannot be expressed as a simple ratio of two integers. They have decimal expansions that never repeat or terminate, such as pi (π) and the square root of 2. Irrational numbers are an important concept in the study of real numbers and their properties, as well as in the context of radicals and rational exponents.
Laws of Exponents: The laws of exponents are a set of rules that govern the manipulation and simplification of expressions involving exponents. These laws are fundamental to understanding and working with radicals and rational exponents, which are crucial topics in college algebra.
Like Radicals: Like radicals are square roots, cube roots, or other radical expressions that have the same index and are raised to the same power. They can be combined through addition, subtraction, or multiplication to simplify radical expressions.
Nth Root: The nth root of a number is the value that, when raised to the power of n, equals the original number. It represents the inverse operation of raising a number to a power and is a fundamental concept in the study of radicals and rational exponents.
Perfect Squares: A perfect square is a number that can be expressed as the product of two equal integers. In other words, a perfect square is the result of multiplying a number by itself. These numbers hold important significance in the context of radicals and rational exponents.
Power Rule: The power rule is a fundamental concept in calculus that describes how to differentiate functions raised to a power. It provides a straightforward method for finding the derivative of expressions involving exponents and powers.
Power rule for logarithms: The power rule for logarithms states that the logarithm of a number raised to an exponent is equal to the exponent times the logarithm of the number. Mathematically, $\log_b(a^c) = c \cdot \log_b(a)$ where $b$ is the base.
Principal nth root: The principal nth root of a number $a$ is the unique real number $b$ such that $b^n = a$, where $n$ is a positive integer. When $n$ is even, the principal nth root is non-negative.
Principal square root: The principal square root of a non-negative number is its non-negative square root. It is denoted as $\sqrt{x}$ where $x$ is the number.
Product Property of Radicals: The product property of radicals states that the square root of a product is equal to the product of the square roots. This can be expressed as $$\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$$ for non-negative numbers a and b. This property is essential for simplifying radical expressions and solving equations involving square roots and higher-order roots, making it a fundamental concept in understanding how to work with radicals and rational exponents.
Product Rule: The product rule is a fundamental concept in mathematics that describes the derivative of a product of two functions. It states that the derivative of a product is equal to the product of the derivative of the first function and the second function, plus the product of the first function and the derivative of the second function.
Product rule for logarithms: The product rule for logarithms states that the logarithm of a product is equal to the sum of the logarithms of its factors. Mathematically, $\log_b(xy) = \log_b(x) + \log_b(y)$.
Quotient Property of Radicals: The quotient property of radicals states that the square root of a fraction is equal to the square root of the numerator divided by the square root of the denominator. This property allows for simplifying radical expressions involving division.
Quotient Rule: The quotient rule is a fundamental mathematical concept that describes how to differentiate the ratio or quotient of two functions. It is a crucial tool in the study of calculus and is applicable across various mathematical domains, including exponents, radicals, logarithmic functions, and more.
Quotient rule for logarithms: The quotient rule for logarithms states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator: $\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N$. It simplifies complex expressions involving division inside a logarithm.
Radical: A radical is an expression that uses a root, such as the square root or cube root, to indicate a value that, when raised to a specified power, yields the original number. Radicals are often used to simplify expressions and solve equations involving roots.
Radical: A radical is a mathematical expression that represents a quantity under a root symbol, such as the square root, cube root, or nth root. Radicals are used to represent irrational numbers and are an essential concept in algebra and higher-level mathematics.
Radical expression: A radical expression is an algebraic expression that includes a square root, cube root, or any higher-order root. It involves the radical symbol (√) and can include variables and constants.
Radical expressions: Radical expressions are algebraic expressions that include a root symbol, such as square roots, cube roots, and higher-order roots. They can often be simplified or manipulated using properties of exponents and radicals.
Radical Expressions: A radical expression is a mathematical expression that contains a radical symbol, such as the square root or nth root, which represents a value raised to a fractional power. Radical expressions are fundamental to understanding topics like radicals and rational exponents.
Radical Symbol (√): The radical symbol, also known as the square root symbol, is a mathematical symbol used to represent the square root of a number or expression. It is a fundamental concept in the study of radicals and rational exponents, which are essential topics in college algebra.
Radicand: A radicand is the number or expression inside a radical symbol. It is the value that you want to find the root of.
Radicand: The radicand is the quantity or expression that is placed under a radical sign, such as the square root or cube root. It represents the value or number that is being operated on by the radical function.
Rational Exponent: A rational exponent is an exponent that can be expressed as a fraction, where the numerator is an integer and the denominator is a positive integer. Rational exponents are used to bridge the gap between radicals and integer exponents, providing a more general and flexible way to represent and manipulate powers.
Rational Exponent Theorem: The rational exponent theorem is a fundamental concept in algebra that allows for the simplification and evaluation of expressions involving fractional exponents. It establishes the relationship between radicals and rational exponents, providing a unified way to represent and manipulate these types of expressions.
Rationalizing Denominators: Rationalizing denominators is the process of eliminating radical expressions or fractional exponents from the denominator of a fraction. This is done to simplify the expression and make it more manageable to work with.
Simplifying Radicals: Simplifying radicals refers to the process of reducing the complexity of square root expressions by factoring out perfect squares or cube roots from the radicand. This technique is essential in the context of radicals and rational exponents, as it allows for the manipulation and evaluation of radical expressions.
Square Root: The square root of a number is the value that, when multiplied by itself, produces the original number. It is denoted by the radical symbol, $\sqrt{}$, and represents the inverse operation of squaring a number.
Unlike Radicals: Unlike radicals are square roots or nth roots that have different indices or different radicands. They cannot be combined or simplified together without first converting them to like radicals by finding a common index and a common radicand.
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