Written by the Fiveable Content Team • Last updated August 2025
Written by the Fiveable Content Team • Last updated August 2025
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.
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.
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).
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).
Perfect powers can be used to simplify radical expressions by rewriting them as a product of a perfect power and a smaller radical.
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.