College Algebra

Unit 1 Overview: Prerequisites

1.1 Real Numbers: Algebra Essentials

1.2 Exponents and Scientific Notation

1.3 Radicals and Rational Exponents

1.4 Polynomials

1.5 Factoring Polynomials

1.6 Rational Expressions

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1.2 Exponents and Scientific Notation

Last Updated on June 24, 2024

Exponents are powerful tools that simplify complex calculations. They help us work with very large or small numbers efficiently. Understanding exponent rules is key to mastering algebraic operations and solving real-world problems.

Scientific notation takes exponents further, allowing us to express extreme values concisely. This format is crucial in fields like physics and astronomy, where numbers can be mind-bogglingly large or small. Mastering these concepts opens doors to advanced math and science applications.

Exponent Rules and Properties

Exponent rules for operations

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Product rule multiplies bases with same exponent adds exponents together ( $x^a \cdot x^b = x^{a+b}$ $x^{a} \cdot x^{b} = x^{a + b}$ )
- Combines like bases by adding their exponents ( $2^3 \cdot 2^4 = 2^{3+4} = 2^7$ )
Quotient rule divides bases with same exponent subtracts exponents ( $\frac{x^a}{x^b} = x^{a-b}$ $\frac{x ^{a}}{x ^{b}} = x^{a - b}$ )
- Simplifies fractions by subtracting exponents of like bases ( $\frac{3^5}{3^2} = 3^{5-2} = 3^3$ )
Power rule raises base and exponent to a power multiplies exponents together ( $(x^a)^b = x^{a \cdot b}$ $(x^{a})^{b} = x^{a \cdot b}$ )
- Simplifies expressions with repeated exponents ( $({y^2})^3 = y^{2 \cdot 3} = y^6$ )
- Can also be expressed using index notation: $(x^a)^b = x^{ab}$

Zero and negative exponents

Zero exponent rule any non-zero base raised to 0 equals 1 ( $x^0 = 1$ $x^{0} = 1$ for $x \neq 0$ $x \neq = 0$ )
- Simplifies expressions with zero exponents ( $4^0 = 1$ , $(\frac{2}{3})^0 = 1$ )
Negative exponent rule reciprocal of base raised to positive exponent ( $x^{-a} = \frac{1}{x^a}$ $x^{- a} = \frac{1}{x ^{a}}$ for $x \neq 0$ $x \neq = 0$ )
- Converts negative exponents to positive by taking reciprocal ( $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$ )
Combining negative exponents with other rules
- Simplifies complex expressions ( $\frac{3^{-2}}{4^{-1}} = \frac{\frac{1}{3^2}}{\frac{1}{4^1}} = \frac{1}{9} \cdot 4 = \frac{4}{9}$ )

Powers of products and quotients

Power of a product each factor raised to the power ( $({xy})^n = x^n \cdot y^n$ $(x y)^{n} = x^{n} \cdot y^{n}$ )
- Distributes exponent to each factor ( $(3a)^2 = 3^2 \cdot a^2 = 9a^2$ )
Power of a quotient numerator and denominator each raised to the power ( $(\frac{x}{y})^n = \frac{x^n}{y^n}$ $(\frac{x}{y})^{n} = \frac{x ^{n}}{y ^{n}}$ )
- Applies exponent to numerator and denominator separately ( $(\frac{2}{b})^3 = \frac{2^3}{b^3} = \frac{8}{b^3}$ )

Complex exponential expressions

Combine exponent rules to simplify expressions
- Break down complex expressions step-by-step ( $(4x^2y^{-3})^3 \cdot (2x^{-1}y^2)^{-2}$ $(4 x^{2} y^{- 3})^{3} \cdot (2 x^{- 1} y^{2})^{- 2}$ )
  1. Apply power rule $(4x^2y^{-3})^3 = 4^3 \cdot x^{2 \cdot 3} \cdot y^{-3 \cdot 3} = 64x^6y^{-9}$
  2. Apply negative exponent rule $(2x^{-1}y^2)^{-2} = \frac{1}{2^2x^{-1 \cdot -2}y^{2 \cdot 2}} = \frac{1}{4} \cdot x^2 \cdot \frac{1}{y^4}$
  3. Multiply simplified expressions $64x^6y^{-9} \cdot \frac{1}{4} \cdot x^2 \cdot \frac{1}{y^4} = 16x^8y^{-13}$

Standard vs scientific notation

Scientific notation expresses numbers as product of a value between 1 and 10 and a power of 10 ( $a \times 10^n$ $a \times 1 0^{n}$ , $1 \leq |a| < 10$ $1 \leq ∣ a ∣ < 10$ , $n$ $n$ is an integer)
- Converts large numbers to scientific notation ( $5,670,000 = 5.67 \times 10^6$ )
- Converts small numbers to scientific notation ( $0.00092 = 9.2 \times 10^{-4}$ )
Standard form is the conventional way of writing numbers without exponents

Calculations in scientific notation

Multiply numbers in scientific notation multiply values and add exponents ( $(a \times 10^n) \cdot (b \times 10^m) = (a \cdot b) \times 10^{n+m}$ $(a \times 1 0^{n}) \cdot (b \times 1 0^{m}) = (a \cdot b) \times 1 0^{n + m}$ )
- Multiplies values and adds exponents ( $(4 \times 10^2) \cdot (2 \times 10^4) = (4 \cdot 2) \times 10^{2+4} = 8 \times 10^6$ )
Divide numbers in scientific notation divide values and subtract exponents ( $\frac{a \times 10^n}{b \times 10^m} = (\frac{a}{b}) \times 10^{n-m}$ $\frac{a \times 1 0 ^{n}}{b \times 1 0 ^{m}} = (\frac{a}{b}) \times 1 0^{n - m}$ )
- Divides values and subtracts exponents ( $\frac{8 \times 10^5}{2 \times 10^3} = (\frac{8}{2}) \times 10^{5-3} = 4 \times 10^2$ )

Interpreting scientific notation

Compare magnitudes of numbers in scientific notation by examining exponents
- Identifies larger value based on exponent ( $3.6 \times 10^9$ is greater than $7.2 \times 10^7$ because $10^9 > 10^7$ )
Interpret meaning of exponent in real-world contexts
- Relates exponent to number of zeros in standard notation (The mass of the Earth is approximately $5.97 \times 10^{24}$ kg, where the exponent 24 represents the number of zeros)
Consider significant figures when working with scientific notation to maintain precision

Engineering notation is similar to scientific notation but uses powers of 10 that are multiples of 3
Logarithms are the inverse operation of exponentiation and can be used to solve exponential equations

Applications of exponents

Apply exponent rules and scientific notation to solve problems in various fields (physics, chemistry, engineering)
- Example: A bacteria population starts with 100 cells and doubles every 4 hours. How many bacteria will be present after 24 hours?
  - Set up exponential growth equation $P = P_0 \cdot 2^{\frac{t}{4}}$ , where $P_0 = 100$ and $t = 24$
  - Solve for $P$ $P = 100 \cdot 2^{\frac{24}{4}} = 100 \cdot 2^6 = 100 \cdot 64 = 6400$

Key Terms to Review (30)

Change-of-base formula: The change-of-base formula is used to rewrite logarithms in terms of logs of another base, allowing for easier computation. It is commonly written as $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ where $b$ and $c$ are positive real numbers and $c \neq 1$.

Exponential growth: Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value. This results in the function increasing rapidly over time.

Scientific notation: Scientific notation is a way of expressing very large or very small numbers in the form $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer. It simplifies calculations and representation of numbers by using powers of ten.

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, $\log_b(xy) = \log_b(x) + \log_b(y)$.

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.

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.

Standard form: Standard form of a linear equation in one variable is written as $Ax + B = 0$, where $A$ and $B$ are constants and $x$ is the variable. The coefficient $A$ should not be zero.

Caret (^) : The caret (^) is a mathematical symbol that represents exponentiation, indicating that a number or variable is raised to a specific power. It is a crucial symbol in the context of exponents and scientific notation, as it allows for the concise representation of large or small numbers.

Base: The base is a fundamental component in various mathematical concepts, serving as a reference point or starting value. It is a crucial element in understanding exponents, exponential functions, logarithmic functions, and geometric sequences, among other topics.

E Notation: E notation, also known as scientific notation, is a compact way of writing very large or very small numbers. It uses the letter 'E' (or sometimes 'e') to represent the power of 10 by which the number is multiplied, allowing for efficient representation of numbers that would otherwise be cumbersome to write out in full.

Exponential Equations: Exponential equations are mathematical expressions where the unknown variable appears as the exponent. These equations describe situations where a quantity grows or decays at a constant rate over time, and they are commonly used to model real-world phenomena such as population growth, radioactive decay, and compound interest.

Engineering Notation: Engineering notation is a form of scientific notation that uses powers of 10 with exponents that are multiples of 3. It is a way of expressing very large or very small numbers in a more compact and easily readable format, particularly in the field of engineering.

Exponents: Exponents are mathematical notations that represent repeated multiplication of a number by itself. They are used to express very large or very small numbers concisely and to simplify algebraic expressions involving powers of the same base.

Exponential Decay: Exponential decay is a mathematical model that describes the gradual reduction or diminishment of a quantity over time. It is characterized by an initial value that decreases by a constant proportion during each successive time interval, resulting in an exponential decrease. This concept is fundamental to understanding various phenomena in fields such as physics, chemistry, biology, and finance.

Exponential Growth: Exponential growth is a pattern of change where a quantity increases at a rate proportional to its current value. This means the quantity grows by a consistent percentage over equal intervals of time, leading to rapid, accelerating growth. Exponential growth is a fundamental concept in mathematics and has applications across various fields, including biology, economics, and technology.

Index Notation: Index notation, also known as subscript notation, is a way of representing repeated operations or quantities using a subscript. It is commonly used in the context of exponents and scientific notation to concisely express repeated multiplication or division by a specific base or power.

John Napier: John Napier was a 16th century Scottish mathematician, physicist, and astronomer who is best known for his invention of logarithms and the Napier's Bones, which were early mechanical calculators. His contributions had a significant impact on the development of modern mathematics and scientific calculation.

Logarithms: Logarithms are the inverse function of exponents, allowing us to express exponential relationships in a more linear form. They are a powerful mathematical tool that can be used to simplify complex calculations and analyze growth or decay patterns.

Mantissa: The mantissa is the part of a logarithm that comes after the decimal point. It represents the fractional part of the logarithm and is used in scientific notation to express the significant digits of a number.

Order of Magnitude: The order of magnitude of a number is the nearest power of ten to that number. It provides a rough estimate of the scale or size of a quantity, allowing for easy comparison between vastly different values.

Negative Exponents: Negative exponents represent the reciprocal or inverse of the base number. They indicate that the value should be expressed as a fraction with the base as the numerator and 1 as the denominator, rather than as a whole number or positive exponent.

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.

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.

Power of a Product: The power of a product refers to the exponent or power to which a product of multiple factors is raised. It is a fundamental concept in the topics of exponents and scientific notation, where the power of a product is used to simplify and manipulate expressions involving multiple factors with 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 of a Quotient: The power of a quotient refers to the exponent applied to a fraction or ratio. It describes the combined effect of raising both the numerator and denominator to a specific power, which can be used to simplify and evaluate expressions involving division and exponents.

Standard Form: Standard form is a way of expressing mathematical equations or functions in a specific, organized format. It provides a consistent structure that allows for easier manipulation, comparison, and analysis of these mathematical representations across various topics in algebra and beyond.

Significant Figures: Significant figures refer to the meaningful digits in a numerical value that indicate the precision and accuracy of the measurement. They are used to represent the reliability and uncertainty of a reported quantity, especially in the context of scientific notation and calculations involving exponents.

Scientific Notation: Scientific notation is a way of expressing very large or very small numbers using a compact format that combines a decimal number and a power of 10. It is a useful tool for representing and manipulating numbers with extreme values in a concise and standardized manner.

Zero Exponent: The zero exponent, also known as the identity property of exponents, is a fundamental concept in mathematics that states that any non-zero number raised to the power of zero is equal to one. This property is essential in understanding and working with exponents, particularly in the context of scientific notation and algebraic expressions.

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📈college algebra review

1.2 Exponents and Scientific Notation