Question

decide if each number is a perfect square, a perfect cube, or neither.
perfect square: 49
perfect cube: 1,000
neither: 15
8
2·2·2 = 8

Explanation:

Step1: Analyze 49

A perfect square is a number that can be written as \(n^2\) where \(n\) is an integer. We check if there's an integer \(n\) such that \(n^2 = 49\). We know that \(7\times7 = 7^2=49\), so 49 is a perfect square.

Step2: Analyze 1000

A perfect cube is a number that can be written as \(m^3\) where \(m\) is an integer. We check if there's an integer \(m\) such that \(m^3 = 1000\). We know that \(10\times10\times10=10^3 = 1000\), so 1000 is a perfect cube.

Step3: Analyze 15

We check if 15 is a perfect square: we test integers. \(3^2 = 9\), \(4^2=16\), so there's no integer \(n\) with \(n^2 = 15\). We check if 15 is a perfect cube: \(2^3 = 8\), \(3^3=27\), so there's no integer \(m\) with \(m^3=15\). So 15 is neither.

Step4: Analyze 8

A perfect cube is a number that can be written as \(k^3\) where \(k\) is an integer. We check if there's an integer \(k\) such that \(k^3=8\). We know that \(2\times2\times2 = 2^3=8\), so 8 is a perfect cube.

Answer:

• 49: perfect square (since \(7^2 = 49\))
• 1000: perfect cube (since \(10^3=1000\))
• 15: neither (not a square or cube of an integer)
• 8: perfect cube (since \(2^3 = 8\))