Glossary

10.2 Use Multiplication Properties of Exponents

3 min read•june 25, 2024

Exponents are powerful tools in algebra, letting us express repeated concisely. They're key to simplifying complex expressions and solving equations. Understanding properties helps us manipulate expressions efficiently.

We'll explore how to simplify exponents, use the product and properties, and combine multiple exponent rules. We'll also cover multiplication and advanced concepts like and .

Exponent Properties

Simplification of polynomial exponents

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  • Identify the (the or number being multiplied) and exponent (the power the base is raised to) in each term of the polynomial expression
  • Combine by adding the coefficients (the numbers in front of the variables) while keeping the same base and exponent (5x2−2x2=3x25x^2 - 2x^2 = 3x^2)
  • Simplify any terms with exponents using the appropriate exponent properties such as the product property, power property, or product to a power property
  • Apply the to expand expressions with exponents (e.g., 2(x3+y3)=2x3+2y32(x^3 + y^3) = 2x^3 + 2y^3)

Product property for like bases

  • Multiply terms with the same base by keeping the base and adding the exponents according to the product property of exponents: am⋅[an](https://www.fiveableKeyTerm:an)=am+na^m \cdot [a^n](https://www.fiveableKeyTerm:a^n) = a^{m+n}
  • x5⋅x3=x5+3=x8x^5 \cdot x^3 = x^{5+3} = x^8 demonstrates multiplying terms with the same base xx
  • Apply the product property to any real number base, including variables, constants, or complex expressions

Power property in exponents

  • Raise a power to another power by keeping the base and multiplying the exponents according to the power property of exponents: (am)n=am⋅n(a^m)^n = a^{m \cdot n}
  • (y3)4=y3⋅4=y12(y^3)^4 = y^{3 \cdot 4} = y^{12} shows raising a power y3y^3 to another power 44
  • Use parentheses to clearly indicate the base being raised to a power to avoid confusion or errors

Product to power property

  • Raise each factor in a product to a power according to the product to a power property: (a⋅b)n=an⋅bn(a \cdot b)^n = a^n \cdot b^n
  • (3x)2=32⋅x2=9x2(3x)^2 = 3^2 \cdot x^2 = 9x^2 demonstrates raising a product 3x3x to the power 22
  • Extend the product to a power property to products with more than two factors by raising each factor to the power

Combining multiple exponent properties

  • Break down complex expressions into smaller parts and identify the appropriate exponent property to apply to each part
  • Multiply terms with the same base by adding exponents (product property), raise a power to another power by multiplying exponents (power property), or raise each factor in a product to a power (product to a power property)
  • Simplify the resulting expression by combining like terms and applying the (2x3(4x2)2=2x3⋅16x4=32x72x^3(4x^2)^2 = 2x^3 \cdot 16x^4 = 32x^7)

Monomial multiplication with exponents

  • Multiply the coefficients of the monomials (terms with a single variable and exponent) and apply the product property of exponents for each variable
  • Add the exponents of like variables when multiplying monomials (2x3y⋅5xy2=10x4y32x^3y \cdot 5xy^2 = 10x^4y^3)
  • Simplify the resulting monomial by combining the and variables with their respective exponents

Applying Exponent Properties

Simplification of polynomial exponents

  • Combine like terms in polynomial expressions by adding coefficients of terms with the same variables and exponents (2x2+3x−x2+4x=x2+7x2x^2 + 3x - x^2 + 4x = x^2 + 7x)
  • Simplify polynomials by applying exponent properties to each term and combining like terms
  • Factor polynomials when possible to represent the expression as a product of its factors (x2−9=(x+3)(x−3)x^2 - 9 = (x+3)(x-3))

Advanced Exponent Concepts

  • Use scientific notation to represent very large or very small numbers using exponents (e.g., 6.02×10236.02 \times 10^{23})
  • Understand exponential expressions as a way to represent repeated multiplication (e.g., 25=2×2×2×2×22^5 = 2 \times 2 \times 2 \times 2 \times 2)
  • Apply rational exponents to express roots and fractional powers (e.g., x12=xx^{\frac{1}{2}} = \sqrt{x}, x34=x34x^{\frac{3}{4}} = \sqrt[4]{x^3})

Key Terms to Review (22)

A^n: The term $a^n$ represents the exponent notation, where $a$ is the base and $n$ is the exponent. This notation is used to express repeated multiplication of the base $a$ by itself $n$ times.
Base: In mathematics, the term 'base' refers to the fundamental unit or quantity from which other values or quantities are derived. It serves as a reference point or starting point for various mathematical concepts and operations across different areas of study.
Coefficient: A coefficient is a numerical factor that multiplies a variable in an algebraic expression. It represents the scale or magnitude of the variable, indicating how much of that variable is present in the expression.
Distributive Property: The distributive property is a fundamental algebraic rule that states the product of a number and a sum is equal to the sum of the individual products. It allows for the simplification of expressions by distributing a factor across multiple terms within a parenthesis or other grouping symbol.
Division: Division is a fundamental mathematical operation that involves partitioning a quantity into equal parts or groups. It represents the inverse of multiplication, allowing us to find how many times one number is contained within another. This key term is essential in understanding various mathematical concepts, from whole numbers to exponents and scientific notation.
Exponent: An exponent is a mathematical symbol that indicates the number of times a base number is multiplied by itself. It represents repeated multiplication and is used to express large numbers concisely. Exponents are a fundamental concept in algebra and are crucial for understanding and working with expressions, polynomials, and scientific notation.
Exponential Expression: An exponential expression is a mathematical expression that involves raising a number or variable to a power. It represents repeated multiplication of a base number by itself a specified number of times, known as the exponent.
Like Terms: Like terms are algebraic expressions that have the same variable(s) raised to the same power. They can be combined by adding or subtracting their coefficients, allowing for the simplification of algebraic expressions.
Monomial: A monomial is a single algebraic expression consisting of a single term, which can include variables, coefficients, and exponents. Monomials are the building blocks of polynomials, which are expressions made up of two or more monomials.
Multiplication: Multiplication is a mathematical operation that involves the repeated addition of a number to itself. It is one of the four basic arithmetic operations, along with addition, subtraction, and division. Multiplication is used to find the total number of items or the area of a rectangle, and it is a fundamental concept in various mathematical contexts, including algebra, geometry, and statistics.
Negative Exponent Rule: The negative exponent rule states that for any nonzero base $a$ and any integer exponent $n$, $a^{-n} = \frac{1}{a^n}$. This rule allows for the simplification of expressions involving negative exponents by rewriting them as positive exponents with a reciprocal base.
Order of Operations: The order of operations is a set of rules that dictate the sequence in which mathematical operations should be performed to evaluate an expression. This term is crucial in the context of evaluating, simplifying, and translating expressions, as well as solving equations using various properties of equality.
Polynomial: A polynomial is an algebraic expression that consists of variables and coefficients, combined using the operations of addition, subtraction, multiplication, and non-negative integer exponents. Polynomials are fundamental to understanding and working with algebraic expressions, as they form the building blocks for many mathematical concepts and applications.
Power: Power is the rate at which work is done or energy is transferred. It is the measure of the strength or force behind an action and is a fundamental concept in mathematics, physics, and various scientific disciplines.
Power of a Power: The power of a power is a mathematical operation where an exponent is raised to another exponent. This concept is particularly relevant in the context of understanding and applying the multiplication properties of exponents, as it allows for the simplification and manipulation of complex exponential expressions.
Power of a Product: The power of a product refers to the exponent or power that is applied to the product of multiple factors. It describes the relationship between the exponents of the individual factors and the exponent of the entire product expression.
Product of Powers: The product of powers is a rule in exponent arithmetic that states the product of two numbers with the same base can be expressed by adding the exponents. This rule simplifies the multiplication of numbers with the same base by combining the exponents.
Rational Exponents: Rational exponents are a way of expressing fractional or negative exponents using a rational number, such as a fraction or a decimal. They provide a consistent way to represent and evaluate expressions involving powers with non-integer exponents.
Scientific Notation: Scientific notation is a concise way of expressing very large or very small numbers by representing them as a product of a number between 1 and 10 and a power of 10. This format allows for more efficient handling and manipulation of such numbers.
Variable: A variable is a symbol, typically a letter, that represents an unknown or changeable quantity in an algebraic expression or equation. It is a fundamental concept in algebra that allows for the generalization of mathematical relationships and the solution of problems involving unknown values.
X^m: The term $x^m$ represents the mathematical expression where $x$ is a variable and $m$ is the exponent. It denotes the operation of raising $x$ to the power of $m$, which results in multiplying $x$ by itself $m$ times.
Zero Exponent Rule: The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. This rule is an important concept in the context of using multiplication properties of exponents and dividing monomials.
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