Simplify.
$\sqrt{27}$
$\sqrt{27}$
Simplify.
$\sqrt{27}$
To simplify the expression \(\sqrt{27}\), we aim to write it in the form \(a\sqrt{b}\), where \(a\) and \(b\) are integers and the radicand \(b\) is as small as possible. This involves identifying the largest perfect square that divides the number 27. A perfect square is an integer that is the square of another integer, such as \(1, 4, 9, 16, \dots\).
First, we find the prime factorization of 27. Since 27 is divisible by 3, we have:
\[27 = 3 \times 9 = 3 \times 3 \times 3 = 3^3\]
Looking at the factors, we identify the largest perfect square factor of 27. The factors are 1, 3, 9, and 27. Among these, the largest perfect square factor is 9, since \(9 = 3^2\).
Using the property of square roots that \(\sqrt{u \cdot v} = \sqrt{u} \cdot \sqrt{v}\) for any non-negative real numbers \(u\) and \(v\), we can rewrite the expression as:
\[\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \cdot \sqrt{3}\]
Since the square root of 9 is 3, the expression simplifies to:
\[3 \cdot \sqrt{3} = 3\sqrt{3}\]
Because 3 has no square factors other than 1, this is the simplest form.
In real life, radicals often appear when dealing with geometry and physics. For example, if you have a square plot of land with an area of \(27\) square meters, calculating the side length of the plot requires you to find \(\sqrt{27}\). By simplifying this to \(3\sqrt{3}\) meters, you can provide a more intuitive and exact measurement, which is approximately \(3 \times 1.732 \approx 5.196\) meters. This makes it easier to work with building materials and plan your layout accurately!