Question
a certain number is both a perfect square and a perfect cube root always be different numbers explain
Answers
GPT-4o mini
A number that is both a perfect square and a perfect cube is known as a perfect sixth power. This means it can be expressed in the form \( n^6 \) for some integer \( n \).
**Explanation:**
1. **Perfect Square Definition**: A number is a perfect square if it can be expressed as \( a^2 \), where \( a \) is an integer. This means it can be represented in the prime factorization as all prime factors raised to even powers.
2. **Perfect Cube Definition**: A number is a perfect cube if it can be expressed as \( b^3 \), where \( b \) is also an integer. This means it can be represented in the prime factorization as all prime factors raised to powers that are multiples of three.
3. **Perfect Sixth Power**: For a number to be both a perfect square and a perfect cube, it must have prime factors raised to powers that are both even and multiples of three. The smallest common multiple of 2 (for squares) and 3 (for cubes) is 6. Therefore, we express the number as \( c^6 \) for some integer \( c \).
4. **Different Values**: Taking different integers for \( n \) will yield different perfect sixth powers. For example:
- If \( n = 1 \), then \( 1^6 = 1 \).
- If \( n = 2 \), then \( 2^6 = 64 \).
- If \( n = 3 \), then \( 3^6 = 729 \), and so forth.
Each \( n^6 \) produces a distinct number, demonstrating that while a number can be uniquely both a perfect square and a perfect cube, different integers \( n \) produce different sixth powers.
In summary, numbers that are both perfect squares and perfect cubes are indeed special cases known as perfect sixth powers, and the corresponding sixth powers of distinct integers are always different from one another.
**Explanation:**
1. **Perfect Square Definition**: A number is a perfect square if it can be expressed as \( a^2 \), where \( a \) is an integer. This means it can be represented in the prime factorization as all prime factors raised to even powers.
2. **Perfect Cube Definition**: A number is a perfect cube if it can be expressed as \( b^3 \), where \( b \) is also an integer. This means it can be represented in the prime factorization as all prime factors raised to powers that are multiples of three.
3. **Perfect Sixth Power**: For a number to be both a perfect square and a perfect cube, it must have prime factors raised to powers that are both even and multiples of three. The smallest common multiple of 2 (for squares) and 3 (for cubes) is 6. Therefore, we express the number as \( c^6 \) for some integer \( c \).
4. **Different Values**: Taking different integers for \( n \) will yield different perfect sixth powers. For example:
- If \( n = 1 \), then \( 1^6 = 1 \).
- If \( n = 2 \), then \( 2^6 = 64 \).
- If \( n = 3 \), then \( 3^6 = 729 \), and so forth.
Each \( n^6 \) produces a distinct number, demonstrating that while a number can be uniquely both a perfect square and a perfect cube, different integers \( n \) produce different sixth powers.
In summary, numbers that are both perfect squares and perfect cubes are indeed special cases known as perfect sixth powers, and the corresponding sixth powers of distinct integers are always different from one another.