Question

19.) a= 3√5, b= 6, c= ???

Explanation:

Assuming this is a right - triangle problem with \(a\) and \(b\) as legs and \(c\) as the hypotenuse, we use the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\).

Step 1: Calculate \(a^{2}\)

Given \(a = 3\sqrt{5}\), then \(a^{2}=(3\sqrt{5})^{2}\). According to the property \((ab)^{n}=a^{n}b^{n}\) and \((\sqrt{m})^{2}=m\) (\(m\geq0\)), we have \((3\sqrt{5})^{2}=3^{2}\times(\sqrt{5})^{2}=9\times5 = 45\).

Step 2: Calculate \(b^{2}\)

Given \(b = 6\), then \(b^{2}=6^{2}=36\).

Step 3: Calculate \(a^{2}+b^{2}\)

Add the results of \(a^{2}\) and \(b^{2}\) together: \(a^{2}+b^{2}=45 + 36=81\).

Step 4: Calculate \(c\)

Since \(c=\sqrt{a^{2}+b^{2}}\) and \(a^{2}+b^{2}=81\), then \(c=\sqrt{81}=9\).

Answer:

\(c = 9\)