Simplify.
$5 \sqrt{147}$
$5 \sqrt{147}$
Simplify.
$5 \sqrt{147}$
To simplify the expression \(5\sqrt{147}\), we first analyze the radicand, which is 147. Simplifying a radical involves identifying any perfect square factors within the radicand. A perfect square is an integer that is the square of another integer, such as \(1, 4, 9, 16, 25, 36, 49, \dots\). By factoring 147, we look for these perfect square factors to pull them outside the square root sign.
We factorize 147 to identify its components:
\[147 = 3 \times 49\]
Since 49 is a perfect square, as \(49 = 7^2\), we can rewrite the radical \(\sqrt{147}\) as:
\[\sqrt{147} = \sqrt{49 \times 3} = \sqrt{49} \times \sqrt{3}\]
Calculating the square root of 49 gives us 7, so the expression simplifies to:
\[\sqrt{147} = 7\sqrt{3}\]
Now, we substitute this back into the original expression:
\[5\sqrt{147} = 5 \times (7\sqrt{3})\]
By multiplying the constants, we get:
\[5 \times 7 = 35\]
Therefore, the simplified form of the expression is \(35\sqrt{3}\).
Simplifying radicals is essential in many fields like architecture and engineering. For example, if you are calculating the length of a support beam using the Pythagorean theorem, you might end up with an expression like \(5\sqrt{147}\) units. Simplifying this to \(35\sqrt{3}\) makes it much easier to estimate the numerical value, as \(\sqrt{3}\) is approximately \(1.732\). Thus, \(35 \times 1.732 \approx 60.62\) units, helping you quickly determine the exact material required for a construction project.