Perfect Powers

Definition

Perfect powers refer to the result of raising a positive integer to a positive integer power. They represent the product of a number multiplied by itself a certain number of times, where both the base and exponent are integers greater than 1.

5 Must Know Facts For Your Next Test

1. Perfect powers can be expressed in the form $a^n$, where $a$ is the base and $n$ is the exponent, and both $a$ and $n$ are positive integers greater than 1.
2. The square of a number is a perfect power, where the exponent is 2, such as 4 (2^2), 9 (3^2), and 16 (4^2).
3. The cube of a number is a perfect power, where the exponent is 3, such as 8 (2^3), 27 (3^3), and 64 (4^3).
4. Perfect powers can be used to simplify radical expressions by rewriting them as a product of a perfect power and a smaller radical.
5. Understanding perfect powers is crucial for working with higher roots, as they can be used to identify and manipulate the radicands in these expressions.

Review Questions

• Explain how perfect powers can be used to simplify radical expressions.
  • Perfect powers can be used to simplify radical expressions by rewriting them as a product of a perfect power and a smaller radical. For example, the expression $\sqrt{64}$ can be simplified to $8$, as 64 is a perfect power (4^2). Similarly, $\sqrt{81}$ can be simplified to $3\sqrt{3}$, as 81 is a perfect power (3^4).
• Describe the relationship between perfect powers and higher roots.
  • Understanding perfect powers is crucial for working with higher roots, as they can be used to identify and manipulate the radicands in these expressions. When the radicand of a higher root is a perfect power, it can be rewritten as a product of a perfect power and a smaller radical, making the expression easier to simplify. For instance, $\sqrt[4]{256}$ can be simplified to $4$, as 256 is a perfect power (4^4).
• Evaluate the expression $\sqrt[6]{729}$ by identifying the perfect power within the radicand.
  • To evaluate the expression $\sqrt[6]{729}$, we can identify the perfect power within the radicand. In this case, 729 is a perfect power, as it can be written as 3^6. Therefore, we can rewrite the expression as $\sqrt[6]{3^6} = 3$. The final answer is 3.

Related terms

Base: The number that is being raised to a power, also known as the radicand.

Exponent: The number that represents how many times the base is multiplied by itself.

Radical: The symbol (√) used to represent the process of finding the root of a number.