Question 10 of 28intermediate🔍 AnalyzeShort Answer3 marks

64 is a square number (82)(8^2) and a cube number (43)(4^3). Are there other numbers that are both squares and cubes? Is there a way to describe such numbers in general?

Correct Answer

Yes, numbers that are both squares and cubes are sixth powers.

Examples:

1=12=131 = 1^2 = 1^3 64=82=4364 = 8^2 = 4^3 729=272=93729 = 27^2 = 9^3 4096=642=1634096 = 64^2 = 16^3

General Rule: A number is a square if its exponent is divisible by 2. A number is a cube if its exponent is divisible by 3. For a number to be both, its exponent must be divisible by both 2 and 3, i.e., by 6. ∴ Numbers of the form n6n^6 are both squares and cubes.

Formula: n6=(n3)2=(n2)3n^6 = (n^3)^2 = (n^2)^3

Exercise: Figure it Out (Section 2) | Q: 10 | (Page Page 27)
For More Understanding

Explanation

A number is a square if its exponent is divisible by 2, and a cube if its exponent is divisible by 3. For a number to be both, its exponent must be divisible by both 2 and 3, i.e., by 6. So numbers of the form n6n^6 are both squares (n6=(n3)2n^6 = (n^3)^2) and cubes (n6=(n2)3n^6 = (n^2)^3).

Solution Steps

  1. A number is a square if its exponent is divisible by 2.

  2. A number is a cube if its exponent is divisible by 3.

  3. For a number to be both, its exponent must be divisible by both 2 and 3, i.e., by 6.

  4. Numbers of the form n6n^6 are both squares (n6=(n3)2n^6 = (n^3)^2) and cubes (n6=(n2)3n^6 = (n^2)^3).

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