- The expression \(15^2 = \boxed{225}\) and \(\sqrt{225} = \boxed{15}\) are inverse operations.
- Similar inverse relationships hold for \(11^2 = \boxed{121}\) and \(\sqrt{121} = \boxed{11}\).
- Arithmetic with radicals like \(\sqrt{16} + \sqrt{9} = \boxed{7}\) and \(\sqrt{16+9} = \boxed{5}\) shows that square roots are not distributive over addition.
- Decimal square roots like \(\sqrt{0.25} = \boxed{0.5}\) and \(\sqrt{0.81} = \boxed{0.9}\) follow the same principles as integers.
Explanation
- Analysis of Expressions
The objective is to simplify various mathematical expressions involving exponents, square roots, and basic arithmetic. Each expression is evaluated independently to verify its correctness. The list includes:
- \(15^2 = 225\)
- \(\sqrt{225} = 15\)
- \(11^2 = 121\)
- \(\sqrt{121} = 11\)
- \(\sqrt{16} + \sqrt{9} = 4 + 3 = 7\)
- \(\sqrt{36} + \sqrt{64} = 6 + 8 = 14\)
- \(\sqrt{25} - \sqrt{9} = 5 - 3 = 2\)
- \(\sqrt{16 + 9} = \sqrt{25} = 5\)
- \(\sqrt{36 + 64} = \sqrt{100} = 10\)
- \(\sqrt{25 - 9} = \sqrt{16} = 4\)
- \(\sqrt{0.25} = 0.5\)
- \(\sqrt{0.81} = 0.9\)
- \(\sqrt{0.01} = 0.1\)
- Step-by-Step Evaluation
Now, let's look at the evaluations:
- \(15^2\) is \(15 \times 15 = 225\).
- The square root of 225, \(\sqrt{225}\), is the number that, when squared, equals 225. Since \(15^2 = 225\), \(\sqrt{225} = 15\).
- \(11^2\) is \(11 \times 11 = 121\).
- \(\sqrt{121} = 11\).
- For \(\sqrt{16} + \sqrt{9}\), we compute \(\sqrt{16} = 4\) and \(\sqrt{9} = 3\), giving \(4 + 3 = 7\).
- For \(\sqrt{36} + \sqrt{64}\), we compute \(\sqrt{36} = 6\) and \(\sqrt{64} = 8\), giving \(6 + 8 = 14\).
- For \(\sqrt{25} - \sqrt{9}\), we compute \(\sqrt{25} = 5\) and \(\sqrt{9} = 3\), giving \(5 - 3 = 2\).
- For \(\sqrt{16 + 9}\), we first add the numbers inside the root: \(\sqrt{25} = 5\).
- For \(\sqrt{36 + 64}\), we compute \(\sqrt{100} = 10\).
- For \(\sqrt{25 - 9}\), we compute \(\sqrt{16} = 4\).
- For decimal square roots: \(\sqrt{0.25} = 0.5\) since \(0.5 \times 0.5 = 0.25\); \(\sqrt{0.81} = 0.9\) since \(0.9 \times 0.9 = 0.81\); and \(\sqrt{0.01} = 0.1\) since \(0.1 \times 0.1 = 0.01\). All calculations are consistent with the provided values.
- Conclusion and Results
All expressions have been verified and their simplified values are confirmed as correct.
- \(15^2 = \boxed{225}\)
- \(\sqrt{225} = \boxed{15}\)
- \(11^2 = \boxed{121}\)
- \(\sqrt{121} = \boxed{11}\)
- \(\sqrt{16} + \sqrt{9} = \boxed{7}\)
- \(\sqrt{36} + \sqrt{64} = \boxed{14}\)
- \(\sqrt{25} - \sqrt{9} = \boxed{2}\)
- \(\sqrt{16 + 9} = \boxed{5}\)
- \(\sqrt{36 + 64} = \boxed{10}\)
- \(\sqrt{25 - 9} = \boxed{4}\)
- \(\sqrt{0.25} = \boxed{0.5}\)
- \(\sqrt{0.81} = \boxed{0.9}\)
- \(\sqrt{0.01} = \boxed{0.1}\)
Examples
Square roots and squares are essential in everyday life. For instance, when designing a square garden, if you know the total area of \(144 \text{ m}^2\), you use the square root to find the side length (\(\sqrt{144} = 12 \text{ m}\)). Similarly, if you want to calculate the total floor space of a square room with side length \(15 \text{ m}\), you square the length (\(15^2 = 225 \text{ m}^2\)). Understanding these operations helps in construction, interior design, and even simple household projects.