UpStudy Free Solution:

To simplify \(\sqrt { 50} \), you need to factor the number inside the square root into its prime factors and then simplify.

First, factor 50 into its prime factors:

\[50 = 2 \times 25\]

\[25 = 5 \times 5\]

So,

\[50 = 2 \times 5^ 2\]

Now, apply the square root to each factor:

\[\sqrt { 50} = \sqrt { 2 \times 5^ 2} \]

Since \(\sqrt { a \times b} = \sqrt { a} \times \sqrt { b} \), we get:

\[\sqrt { 50} = \sqrt { 2} \times \sqrt { 5^ 2} \]

The square root of \(5^ 2\) is 5:

\[\sqrt { 50} = \sqrt { 2} \times 5\]

Therefore, the simplest radical form of \(\sqrt { 50} \) is:

\[5\sqrt { 2} \]

So, \(\sqrt { 50} \) in simplest radical form is \(5\sqrt { 2} \).

Supplemental Knowledge

Simplifying radicals is a key skill in algebra that involves expressing a square root in its simplest form. This process often requires factoring the number under the square root into its prime factors and then simplifying.

Here’s a step-by-step guide to simplifying radicals:

1. Prime Factorization: Break down the number inside the square root into its prime factors.

2. Pairing Factors: Identify pairs of prime factors because the square root of a pair of identical numbers is simply one of those numbers.

3. Simplify: Rewrite the expression by taking out pairs from under the square root.

For example, let's simplify \(\sqrt { 72} \):

1. Factorize 72:

\[72 = 2 \times 36 = 2 \times ( 6 \times 6) = 2 \times ( 2 \times 3) \times ( 2 \times 3) \]

So,

\[72 = 2^ 3 \times 3^ 2\]

2. Identify Pairs:

- \(3^ 2\) forms a pair.

- \(2^ 3\) has one pair and one single factor left.

3. Simplify:

- The pair of \(3\) comes out as a single \(3\).

- The pair of \(2\) comes out as a single \(2\).

Thus,

\[\sqrt { 72} = \sqrt { ( 2^ 2) \cdot ( 3^ 2) \cdot 2} = ( 2) ( 3) \sqrt { 2} = 6\sqrt { 2} \]

Understanding radicals is integral for succeeding in algebra and higher level mathematics courses, so if you find yourself stuck or need additional clarification UpStudy is here to assist!
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