Product of Powers Property

Definition

The product of powers property is a rule in exponent arithmetic that states that when two numbers with the same base are multiplied, the exponents can be added together to form the new exponent of the product. This property is particularly useful in the context of integer exponents and scientific notation.

5 Must Know Facts For Your Next Test

1. The product of powers property can be expressed mathematically as: $a^m \cdot a^n = a^{m+n}$, where $a$ is the base and $m$ and $n$ are the exponents.
2. This property allows for the simplification of expressions involving the multiplication of numbers with the same base but different exponents.
3. In the context of scientific notation, the product of powers property can be used to multiply numbers with the same base (10) but different exponents.
4. The product of powers property is a fundamental rule in exponent arithmetic and is essential for working with integer exponents and scientific notation.
5. Understanding and applying the product of powers property is crucial for simplifying and manipulating expressions involving exponents, which is a key skill in elementary algebra.

Review Questions

• Explain how the product of powers property can be used to simplify an expression involving the multiplication of two numbers with the same base but different exponents.
  • The product of powers property states that when two numbers with the same base are multiplied, the exponents can be added together to form the new exponent of the product. For example, if we have the expression $2^3 \cdot 2^4$, we can apply the product of powers property to simplify it as $2^{3+4} = 2^7$. This property allows us to efficiently multiply numbers with the same base by adding their exponents, which is particularly useful when working with integer exponents and scientific notation.
• Describe how the product of powers property can be applied in the context of scientific notation.
  • In scientific notation, numbers are expressed as a product of a decimal value (between 1 and 10) and a power of 10. The product of powers property can be used to multiply numbers in scientific notation by adding the exponents of the 10s. For instance, if we have the expression $(3.2 \times 10^5) \cdot (4.1 \times 10^3)$, we can apply the product of powers property to simplify it as $(3.2 \cdot 4.1) \times 10^{5+3} = 13.12 \times 10^8$. This property is essential for performing operations with numbers in scientific notation.
• Analyze how the product of powers property can be used to evaluate expressions involving integer exponents with the same base.
  • The product of powers property is a fundamental rule in the context of integer exponents. It allows us to evaluate expressions involving the multiplication of numbers with the same base by adding their exponents. For example, if we have the expression $5^2 \cdot 5^3$, we can apply the product of powers property to simplify it as $5^{2+3} = 5^5$. This property enables us to efficiently manipulate and simplify expressions with integer exponents, which is a crucial skill in elementary algebra. Understanding and applying the product of powers property is essential for working with exponents and performing operations with integer exponents and scientific notation.

Related terms

Exponent:

A number that indicates how many times the base number is multiplied by itself.

Base:

The number that is being raised to a power.

Scientific Notation:

A way of writing very large or very small numbers using exponents of 10.