Which is a perfect square?
[tex]$6^1$[/tex]
[tex]$6^2$[/tex]
[tex]$6^3$[/tex]
[tex]$6^5$[/tex]
Answer (1)
A perfect square is a number whose square root is an integer.
Calculate the values: 6 1 = 6 , 6 2 = 36 , 6 3 = 216 , 6 5 = 7776 .
Check which value has an integer square root: 6 ≈ 2.45 , 36 = 6 , 216 ≈ 14.70 , 7776 ≈ 88.18 .
6 2 = 36 is the only perfect square. 6 2
Explanation
Problem Analysis We are asked to identify which of the given numbers is a perfect square. A perfect square is a number that can be obtained by squaring an integer. In other words, a perfect square has an integer square root. The numbers we are given are 6 1 , 6 2 , 6 3 , and 6 5 .
Checking Each Option Let's examine each option:
6 1 = 6 . The square root of 6 is not an integer ( 6 ≈ 2.449 ), so 6 1 is not a perfect square.
6 2 = 36 . The square root of 36 is 6, which is an integer ( 36 = 6 ), so 6 2 is a perfect square.
6 3 = 216 . The square root of 216 is not an integer ( 216 ≈ 14.697 ), so 6 3 is not a perfect square.
6 5 = 7776 . The square root of 7776 is not an integer ( 7776 ≈ 88.182 ), so 6 5 is not a perfect square.
Conclusion Therefore, the only perfect square among the given options is 6 2 = 36 .
Alternative Approach Alternatively, we can consider the exponents. A number in the form a b is a perfect square if b is an even number and a is a perfect square, or if b is even regardless of a . In our case, a = 6 . Since 6 is not a perfect square, we need to check if the exponent b is even. Among the options 6 1 , 6 2 , 6 3 , and 6 5 , only 6 2 has an even exponent. Thus, 6 2 is a perfect square.
Examples
Perfect squares are useful in various real-life situations, such as calculating areas and volumes. For example, if you are designing a square garden with an area of 36 square meters, you need to find the side length of the garden by taking the square root of the area. In this case, the side length would be 36 = 6 meters. Understanding perfect squares helps in quickly determining such dimensions without needing a calculator.
