Question

express in simplest radical form.

$10\sqrt{147} + \sqrt{27}$

Explanation:

Step 1: Simplify the radicals

First, factor the numbers inside the square roots into perfect squares and other factors. For \(\sqrt{147}\), we have \(147 = 49\times3\), and for \(\sqrt{27}\), we have \(27 = 9\times3\).
Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (where \(a\) is a perfect square), we get:
\(\sqrt{147}=\sqrt{49\times3}=\sqrt{49}\times\sqrt{3}=7\sqrt{3}\)
\(\sqrt{27}=\sqrt{9\times3}=\sqrt{9}\times\sqrt{3}=3\sqrt{3}\)

Step 2: Substitute the simplified radicals back into the original expression

The original expression is \(10\sqrt{147}+\sqrt{27}\). Substituting the simplified radicals, we have:
\(10\times7\sqrt{3}+3\sqrt{3}\)

Step 3: Perform the multiplication and combine like terms

First, multiply \(10\) and \(7\): \(10\times7 = 70\), so the expression becomes \(70\sqrt{3}+3\sqrt{3}\).
Since both terms have \(\sqrt{3}\), we can combine them by adding the coefficients: \((70 + 3)\sqrt{3}=73\sqrt{3}\)

Answer:

\(73\sqrt{3}\)