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Questions and Answers
If a is a number and m, n are positive integers, then a^m * a^n = a^______.
m+n
If a is a nonzero number and m, n are positive integers, then a^m / a^n = a^______.
m-n
(a^m)^n = a^______.
mn
(ab)^n = a^n * b^n. For example, (2 * 5)^3 = 2^3 * 5^3 = 8 * 125 = ______.
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In the expression 4^3, the exponent 3 indicates that 4 is multiplied by itself ______ times.
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If a and b are nonzero numbers and n is an integer, then (a/b)^n = ________.
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Negative exponents are used to represent the reciprocal of a base raised to a ________ exponent.
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Fractional exponents, also known as roots, represent the result of raising the base to a ________ power.
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Exponential functions typically take the form y = a^x, where a is the base and x is the ________ variable.
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The behavior of exponential functions is characterized by rapid growth when a > 1 and ________ when 0 < a < 1.
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Study Notes
Exponents: Unlocking Powers of Numbers
Exponents, also known as indices or powers, are a fundamental concept in mathematics that allow us to express and manipulate numbers in specific ways. Let's dive into the world of exponents, beginning with the laws that govern this system.
Definition of an Exponent
An exponent, denoted by the superscript that comes after a number or variable, represents the number of times the base is multiplied by itself. For example, in the expression (4^3), the exponent 3 indicates that 4 is multiplied by itself three times, resulting in (4 \times 4 \times 4 = 64).
Laws of Exponents
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Multiplication property of exponents: If (a) is a number and (m,n) are positive integers, then (a^m \times a^n = a^{m+n}). For instance, (2^3 \times 2^2 = 2^5 = 32).
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Division property of exponents: If (a) is a nonzero number and (m,n) are positive integers, then (a^m \div a^n = a^{m-n}). For example, (\frac{3^5}{3^2} = 3^{5-2} = 3^3 = 27).
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Power of a power property: If (a) is a nonzero number and (m,n) are integers, then ((a^m)^n = a^{mn}). For instance, ((4^2)^3 = 4^{2 \times 3} = 4^6 = 4096).
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Product to a power: If (a) and (b) are nonzero numbers and (n) is an integer, then ((ab)^n = a^n \times b^n). For example, ((2 \times 5)^3 = 2^3 \times 5^3 = 8 \times 125 = 1000).
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Quotient to a power: If (a) and (b) are nonzero numbers and (n) is an integer, then (\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}). For instance, (\left(\frac{5}{3}\right)^3 = \frac{5^3}{3^3} = \frac{125}{27}).
These laws provide a framework for manipulating exponents and simplifying expressions.
Negative Exponents
Negative exponents are used to represent the reciprocal of a base raised to a positive exponent. For example, (a^{-n}) is the same as (\frac{1}{a^n}).
Fractional Exponents
Fractional exponents, also known as roots, represent the result of raising the base to a fractional power. For instance, (a^{\frac{1}{n}}) is the same as the (n^{th}) root of (a).
Exponential Functions
Exponents are closely related to exponential functions, which typically take the form (y = a^x), where (a) is the base and (x) is the independent variable. The behavior of exponential functions is characterized by rapid growth (when (a > 1)) and decay (when (0 < a < 1)).
Exponents and the laws governing them are fundamental tools in mathematics, allowing us to simplify and analyze various expressions and functions. By mastering the laws of exponents, you will lay a firm foundation in algebra, geometry, and calculus.
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Description
Dive into the world of exponents, exploring the definition and laws that govern these powerful mathematical tools. Learn about multiplication, division, power of a power, product to a power, and quotient to a power properties of exponents. Understand how negative and fractional exponents, as well as exponential functions, play a crucial role in algebra and beyond.
