key term - Power Property

Definition

The power property, also known as the power rule, is a fundamental concept in algebra that describes how to simplify and manipulate expressions involving exponents. It allows for the efficient calculation of powers of powers, products of powers, and quotients of powers.

5 Must Know Facts For Your Next Test

1. The power property states that for any positive real numbers $a$ and $b$, and any integers $m$ and $n$, the following holds: $a^m \cdot a^n = a^{m+n}$.
2. The power property can be used to simplify expressions involving products of powers with the same base by adding the exponents.
3. The power property can also be used to simplify expressions involving quotients of powers with the same base by subtracting the exponents.
4. The power property is particularly useful when working with fractional exponents, as it allows for the simplification of complex expressions.
5. Understanding the power property is crucial for solving a wide range of algebraic equations and inequalities, as well as for manipulating and simplifying expressions involving exponents.

Review Questions

• Explain how the power property can be used to simplify an expression involving the product of two powers with the same base.
  • The power property states that for any positive real numbers $a$ and $b$, and any integers $m$ and $n$, the expression $a^m \cdot a^n$ can be simplified to $a^{m+n}$. This is because the power property allows us to add the exponents when multiplying powers with the same base. For example, $3^4 \cdot 3^5$ can be simplified to $3^{4+5} = 3^9$.
• Describe how the power property can be used to simplify an expression involving the quotient of two powers with the same base.
  • The power property also states that for any positive real numbers $a$ and $b$, and any integers $m$ and $n$, the expression $a^m \div a^n$ can be simplified to $a^{m-n}$. This is because the power property allows us to subtract the exponents when dividing powers with the same base. For example, $5^8 \div 5^3$ can be simplified to $5^{8-3} = 5^5$.
• Analyze how the power property can be applied to expressions involving fractional exponents, and explain the significance of this application.
  • The power property is particularly useful when working with expressions that contain fractional exponents. For instance, the expression $\left(a^{1/2}\right)^3$ can be simplified using the power property to $a^{1/2 \cdot 3} = a^{3/2}$. This application of the power property is significant because it allows for the efficient manipulation and simplification of complex expressions involving roots and fractional powers, which are commonly encountered in advanced algebraic problems.

Related terms

Exponent:

A number that represents the power to which a base number is raised, indicating how many times the base is used as a factor.

Base:

The number that is raised to a power, which is multiplied by itself a certain number of times.

Simplification:

The process of reducing an algebraic expression to its simplest form, often by applying various properties and rules.