Simplify.
$\sqrt{90}$
$\sqrt{90}$
Simplify.
$\sqrt{90}$
To simplify the radical \(\sqrt{90}\), we need to find the largest perfect square factor of the radicand, which is 90. A perfect square is an integer that can be expressed as the square of another integer, such as \(1, 4, 9, 16, 25, 36, \dots\). We can do this by examining the prime factorization of 90.
We find the prime factorization of 90:
\[90 = 9 \times 10 = 3^2 \times 2 \times 5\]
The prime factors are 2, 3, and 5. We see that the largest perfect square factor of 90 is \(3^2 = 9\). Any remaining factors, which in this case are 2 and 5, multiply to give 10. Since 10 has no prime factors that are squares (it is square-free), our simplification will be complete once we extract the square root of 9.
Using the property of square roots, \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), we can rewrite the original expression as follows:
\[\sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10}\]
Since \(\sqrt{9} = 3\), the expression simplifies to:
\[3\sqrt{10}\]
This is the simplest radical form because 10 cannot be divided by any perfect square other than 1.
Simplifying radicals is a fundamental skill used in various practical fields such as architecture, construction, and engineering. For example, if you are designing a staircase and need to calculate the diagonal length of a supporting structure with specific height and width dimensions, you might end up with an expression like \(\sqrt{90}\). Simplifying this to \(3\sqrt{10}\) allows for a more precise and manageable numerical approximation (\(3 \times 3.162 \approx 9.486\)) for measuring lengths during construction, ensuring accuracy in your project.