91Ó°ÊÓ

To simplify square roots, we often begin with prime factorization. Prime factorization is the process of breaking down a number into prime numbers that multiply together to give the original number. For example, let's take the number 32, which we need to simplify in our exercise.

The prime factorization of 32 is written as follows:

• 32 is divisible by 2 (the smallest prime number), giving us 16.
  32 = 2 x 16.
• Next, factorize 16: it is also divisible by 2, providing us with 8.
  16 = 2 x 8.
• Continue with 8: divisible by 2, resulting in 4.
  8 = 2 x 4.
• Finally, 4 is divisible by 2 to give us 2 x 2.
  4 = 2 x 2.

Therefore, the prime factorization of 32 is: 32 = 2 x 2 x 2 x 2 x 2 = \(2^5\).

Knowing the prime factors helps us simplify square roots, as seen with \(\sqrt{32}\).

Simplifying radicals involves expressing the square root of a number in its simplest form. First, we factorize the number under the square root into its prime factors, then apply the properties of square roots to break it down further.

Using our example \(\sqrt{32}\):

First, we factorize 32 into its prime factors, \(2^5\). Radicals involving higher powers of a number can be simplified using the rule \(\sqrt{a^m} = a^{(m/2)}\), where 'm' is an even number. Here, we split \(2^5\) into \(2^4 \times 2\), so:

\(\sqrt{32} = \sqrt{2^4 \times 2} = \sqrt{2^4} \times \sqrt{2} = 2^2 \times \sqrt{2} = 4 \sqrt{2}\).

This process involves simplifying the even power separately and multiplying the remaining square roots. Thus, we have simplified \(\sqrt{32}\) to \(\ 4 \sqrt{2}\).

Now let's see how to multiply square roots. The product of square roots can be simplified using the property \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\).

In our exercise, we have two square roots to multiply: \(\ \sqrt{9} \times \sqrt{32}\). Start by simplifying \sqrt{9} and \sqrt{32} separately:

• \(\sqrt{9} = 3 \) since 9 is a perfect square.
• We have already simplified \sqrt{32} to \4 \sqrt{2}.\

Next, multiply these simplified forms:

\(\ 3 \times 4 \sqrt{2} = 12 \sqrt{2}\).

This approach ensures the multiplication of both square roots and their simplified form into one expression, which is easy to understand and work with. This demonstrates how the concept of multiplication of square roots, along with their simplification, results in a straightforward solution. Therefore, the simplified product is \(\12 \sqrt{2}\).