Question

1. list the perfect square numbers from 1 to 100.

1. simplify the following radicals (square roots) by taking out the perfect square numbers.

a) \\(\sqrt{8} =\\) \t\t\t\tb) \\(\sqrt{18} =\\)

c) \\(\sqrt{27} =\\) \t\t\t\td) \\(\sqrt{48} =\\)

Explanation:

Problem 1:

Step1: Recall perfect square definition

A perfect square is \( n^2 \) where \( n \) is an integer. We find \( n \) such that \( 1\leq n^2\leq100 \).

Step2: Find integer values of \( n \)

Solve \( 1\leq n^2\leq100 \). Taking square roots, \( 1\leq n\leq10 \) (since \( n \) is positive integer here, as we consider positive squares; negative \( n \) would give same squares). So \( n = 1,2,\dots,10 \).

Step3: Calculate \( n^2 \)

• \( 1^2 = 1 \)
• \( 2^2 = 4 \)
• \( 3^2 = 9 \)
• \( 4^2 = 16 \)
• \( 5^2 = 25 \)
• \( 6^2 = 36 \)
• \( 7^2 = 49 \)
• \( 8^2 = 64 \)
• \( 9^2 = 81 \)
• \( 10^2 = 100 \)

Step1: Factor 8 into perfect square and other

Factor 8: \( 8 = 4\times2 \), where 4 is a perfect square (\( 2^2 \)).

Step2: Apply square root property

\( \sqrt{8}=\sqrt{4\times2} \). Using \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (for \( a,b\geq0 \)), we get \( \sqrt{4}\times\sqrt{2} \).

Step3: Simplify \( \sqrt{4} \)

\( \sqrt{4}=2 \), so \( \sqrt{8}=2\sqrt{2} \).

Step1: Factor 18 into perfect square and other

Factor 18: \( 18 = 9\times2 \), 9 is \( 3^2 \) (perfect square).

Step2: Apply square root property

\( \sqrt{18}=\sqrt{9\times2}=\sqrt{9}\times\sqrt{2} \).

Step3: Simplify \( \sqrt{9} \)

\( \sqrt{9}=3 \), so \( \sqrt{18}=3\sqrt{2} \).

Answer:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Problem 2a: