Power of a Product

Definition

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.

5 Must Know Facts For Your Next Test

1. The power of a product is calculated by adding the exponents of the individual factors in the product.
2. This property is useful for simplifying expressions involving multiplication of numbers in scientific notation.
3. The power of a product rule states that $(a^m)(a^n) = a^{m+n}$, where $a$ is the base and $m$ and $n$ are the exponents.
4. The power of a product property can be extended to products of more than two factors, allowing for efficient manipulation of complex expressions.
5. Understanding the power of a product is essential for performing operations with scientific notation, which is commonly used in various scientific and mathematical fields.

Review Questions

• Explain the relationship between the power of a product and exponents.
  • The power of a product is directly related to the exponents of the individual factors in the product. Specifically, the power of a product is calculated by adding the exponents of the factors. This property is captured by the rule $(a^m)(a^n) = a^{m+n}$, where $a$ is the base and $m$ and $n$ are the exponents. This relationship allows for the simplification of expressions involving the multiplication of numbers with exponents, which is a crucial skill in working with exponents and scientific notation.
• Describe how the power of a product can be used to manipulate expressions in scientific notation.
  • The power of a product property is particularly useful when working with numbers in scientific notation. In scientific notation, numbers are expressed as the product of a number between 1 and 10 and a power of 10. When multiplying numbers in scientific notation, the power of a product rule can be applied to efficiently combine the exponents of the powers of 10. This allows for the simplification of complex expressions involving multiplication of numbers in scientific notation, which is a common task in various scientific and mathematical contexts.
• Evaluate the expression $(2.5 imes 10^3)(4.8 imes 10^{-2})$ using the power of a product property.
  • To evaluate the expression $(2.5 imes 10^3)(4.8 imes 10^{-2})$ using the power of a product property, we first identify the individual factors and their respective exponents. The first factor is $2.5 imes 10^3$, with an exponent of 3. The second factor is $4.8 imes 10^{-2}$, with an exponent of -2. Applying the power of a product rule, $(a^m)(a^n) = a^{m+n}$, we can simplify the expression as follows: $(2.5 imes 10^3)(4.8 imes 10^{-2}) = (2.5 imes 4.8) imes 10^{3+(-2)} = 12 imes 10^1 = 1.2 imes 10^2$.

Related terms

Exponent:

A mathematical symbol that indicates the power to which a number or expression is to be raised.

Scientific Notation:

A way of writing numbers as the product of a number between 1 and 10 and a power of 10, used to represent very large or very small numbers.

Product: The result of multiplying two or more numbers or quantities.