Is 729 a perfect cube?
Understand the Problem
The question is asking whether the number 729 can be expressed as a cube of an integer. To determine this, we will find the cube root of 729 and see if it yields a whole number.
Answer
The number 729 can be expressed as $9^3$.
Answer for screen readers
The number 729 can be expressed as the cube of the integer 9.
Steps to Solve
- Finding the Cube Root
To determine if 729 can be expressed as a cube of an integer, we need to find its cube root. This can be calculated as:
$$ \sqrt[3]{729} $$
- Calculating the Cube Root
We can simplify the cube root calculation using prime factorization. The first step is to factor 729. We can divide it by 3:
$$ 729 ÷ 3 = 243 $$ $$ 243 ÷ 3 = 81 $$ $$ 81 ÷ 3 = 27 $$ $$ 27 ÷ 3 = 9 $$ $$ 9 ÷ 3 = 3 $$ $$ 3 ÷ 3 = 1 $$
So, the prime factorization of 729 is:
$$ 729 = 3^6 $$
- Expressing as a Cube
To express 729 as a cube, we rewrite it in terms of cubes:
$$ 3^6 = (3^2)^3 = 9^3 $$
Thus, $729 = 9^3$.
The number 729 can be expressed as the cube of the integer 9.
More Information
This means that 729 is a perfect cube, specifically the cube of 9. Perfect cubes are numbers that can be formed by multiplying an integer by itself twice (e.g., $n^3$ where $n$ is an integer).
Tips
- A common mistake is to assume the cube root is a whole number without factoring. Always check the prime factorization to see if you can express the number as a cube.
