Subjects algebra

Square Numbers

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Square Numbers


1. **State the problem:** We are given a 3x3 grid of expressions: \[\sum_{i=2}^3 i, \quad e^3, \quad 6! \\ \int_3^6 x dx, \quad \sqrt{9}, \quad \frac{6\pi}{2} \\ \log_4(25), \quad \frac{3}{9}, \quad \infty\] We need to identify which of these are squares and then list them from least to greatest. 2. **Evaluate each expression:** - \(\sum_{i=2}^3 i = 2 + 3 = 5\) (not a perfect square) - \(e^3 \approx 20.0855\) (not a perfect square) - \(6! = 720\) (not a perfect square) - \(\int_3^6 x dx = \left[\frac{x^2}{2}\right]_3^6 = \frac{6^2}{2} - \frac{3^2}{2} = \frac{36}{2} - \frac{9}{2} = 18 - 4.5 = 13.5\) (not a perfect square) - \(\sqrt{9} = 3\) (this is a perfect square because 3 is the square root of 9, which is a perfect square) - \(\frac{6\pi}{2} = 3\pi \approx 9.4247\) (not a perfect square) - \(\log_4(25)\) (change of base: \(\log_4(25) = \frac{\ln 25}{\ln 4} \approx \frac{3.2189}{1.3863} \approx 2.323\), not a perfect square) - \(\frac{3}{9} = \frac{1}{3} \approx 0.3333\) (not a perfect square) - \(\infty\) (not a number, ignore) 3. **Identify squares:** The only perfect square we have here is from \(\sqrt{9} = 3\) which means 9 is a perfect square (3 squared). The actual number is 3, which itself is not a perfect square, but the expression \(\sqrt{9} = 3\) relates to a perfect square. The question asks to "select all the squares," so the only expression that is a square number or related to a square number is \(\sqrt{9}\). 4. **List from least to greatest:** Only one square value: \(3\). **Final answer:** Only \(\sqrt{9} = 3\) is a perfect square in the grid.