What is $5 \sqrt{28}+\sqrt{63}$ in simplest radical form?
We need to simplify the expression \(5 \sqrt{28} + \sqrt{63}\). To simplify a square root, we look for the largest perfect square factor of the radicand. A radicand is the number inside the square root. Once identified, we can pull the square root of that factor out of the radical using the property \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\).
First, we examine the radicand 28. The factors of 28 are 1, 2, 4, 7, 14, and 28. The largest perfect square factor is 4. We can write 28 as \(4 \times 7\).
Thus, \(\sqrt{28} = \sqrt{4 \cdot 7} = \sqrt{4} \cdot \sqrt{7} = 2\sqrt{7}\).
Now, multiply this by the coefficient 5:
\(5\sqrt{28} = 5 \cdot (2\sqrt{7}) = 10\sqrt{7}\).
Next, we examine the radicand 63. The factors of 63 include 1, 3, 7, 9, 21, and 63. The largest perfect square factor is 9. We can write 63 as \(9 \times 7\).
Thus, \(\sqrt{63} = \sqrt{9 \cdot 7} = \sqrt{9} \cdot \sqrt{7} = 3\sqrt{7}\).
Now that both terms are expressed in terms of \(\sqrt{7}\), we can add them:
\(10\sqrt{7} + 3\sqrt{7} = (10 + 3)\sqrt{7} = 13\sqrt{7}\).
The expression is now in simplest radical form.
Simplifying radicals is a foundational skill in geometry and physics. For example, if you are calculating the diagonal of a rectangular field that measures \(\sqrt{28}\) meters by \(\sqrt{63}\) meters using the Pythagorean theorem, you would need to combine these types of expressions to find the total distance. Understanding how to simplify these terms allows you to work with exact values instead of messy decimals, which keeps your calculations precise in engineering and design projects.