key term - Power Property

Definition

The power property is a fundamental rule in exponent arithmetic that allows for the simplification and manipulation of expressions involving powers. It states that when raising a power to another power, the exponents can be multiplied together.

5 Must Know Facts For Your Next Test

1. The power property states that $a^{m} \cdot a^{n} = a^{m+n}$, where $a$ is the base and $m$ and $n$ are the exponents.
2. This property allows you to combine multiple powers with the same base by adding the exponents together.
3. The power property can be used to simplify expressions involving multiplication of powers with the same base.
4. Applying the power property can help reduce the number of steps required to evaluate or manipulate expressions with exponents.
5. Understanding the power property is essential for mastering exponent arithmetic and solving problems involving multiplication of powers.

Review Questions

• Explain how the power property can be used to simplify an expression involving the multiplication of powers with the same base.
  • The power property states that when you multiply powers with the same base, you can add the exponents together. For example, if you have the expression $a^3 \cdot a^4$, you can use the power property to simplify it to $a^{3+4} = a^7$. This allows you to reduce the number of terms in the expression and make it easier to evaluate.
• Describe a situation where applying the power property would be useful in solving a problem related to exponent arithmetic.
  • Suppose you have an expression like $(a^5)^3$. Using the power property, you can rewrite this as $a^{5 \cdot 3} = a^{15}$. This simplification can be helpful when you need to evaluate the expression or use it in further calculations. The power property allows you to condense multiple powers with the same base into a single power, making the expression more manageable and easier to work with.
• Analyze how the power property relates to the broader concept of exponent arithmetic and its importance in understanding and manipulating expressions involving exponents.
  • The power property is a fundamental rule in exponent arithmetic that underpins our understanding and manipulation of expressions with exponents. It allows us to simplify and combine powers with the same base, which is essential for solving a wide range of algebraic problems. Without the power property, working with exponents would be much more complex and challenging. By mastering this property, students can develop a deeper grasp of exponent arithmetic and apply it effectively to solve a variety of mathematical problems involving powers and exponential expressions.

Related terms

Exponent:

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

Base:

The number that is being raised to a power.

Exponent Arithmetic: The set of rules and operations that govern how exponents are calculated and simplified.