Power of a Power Property

Definition

The power of a power property is a rule in exponent arithmetic that states that when raising a power to another power, the exponents are multiplied. This property is fundamental in simplifying and manipulating expressions involving integer exponents and scientific notation.

5 Must Know Facts For Your Next Test

1. The power of a power property states that $(a^m)^n = a^{m \times n}$, where $a$ is the base and $m$ and $n$ are integers.
2. This property allows for the simplification of expressions involving multiple exponents, such as $(x^3)^4 = x^{3 \times 4} = x^{12}$.
3. The power of a power property is particularly useful when working with scientific notation, as it allows for the quick manipulation of exponents.
4. For example, $(5.6 \times 10^{-3})^2 = (5.6)^2 \times (10^{-3})^2 = 31.36 \times 10^{-6} = 3.136 \times 10^{-5}$.
5. Understanding the power of a power property is essential for efficiently simplifying and evaluating expressions involving integer exponents and scientific notation.

Review Questions

• Explain how the power of a power property can be used to simplify expressions involving integer exponents.
  • The power of a power property states that when a power is raised to another power, the exponents are multiplied. This allows you to simplify expressions with multiple exponents by multiplying the exponents together. For example, $(x^3)^4$ can be simplified to $x^{3 \times 4} = x^{12}$. This property is useful for reducing the number of steps required to evaluate expressions with integer exponents.
• Describe how the power of a power property can be applied when working with scientific notation.
  • When working with scientific notation, the power of a power property is particularly helpful. The property states that $(a^m)^n = a^{m \times n}$, which means that the exponents can be multiplied when raising a number in scientific notation to a power. For instance, $(5.6 \times 10^{-3})^2$ can be simplified to $(5.6)^2 \times (10^{-3})^2 = 31.36 \times 10^{-6} = 3.136 \times 10^{-5}$. This allows for efficient manipulation of numbers expressed in scientific notation.
• Analyze how the power of a power property relates to the broader concepts of integer exponents and scientific notation, and explain its importance in simplifying and evaluating expressions involving these topics.
  • The power of a power property is a fundamental rule that connects the concepts of integer exponents and scientific notation. It allows you to simplify expressions by multiplying the exponents when raising a power to another power. This property is essential when working with both integer exponents and scientific notation, as it enables you to efficiently evaluate and manipulate complex expressions. Understanding how to apply the power of a power property is crucial for successfully navigating topics like integer exponents and scientific notation, as it reduces the number of steps required and helps you arrive at the correct solutions. Mastering this property is a key skill for succeeding in algebra and related mathematical disciplines.

Related terms

Integer Exponents: Exponents that are whole numbers, either positive or negative, used to represent repeated multiplication or division.

Scientific Notation: A way of writing very large or very small numbers using exponents of 10 to express the magnitude of the number.

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