Problem 11 - Prealgebra

Question

Simplify each square root, then combine if possible. Assume all variables represent positive numbers. $$\sqrt{18}+\sqrt{32}$$

Step-by-Step Solution

Answer

The expression simplifies to \(7\sqrt{2}\).

Step 1: Simplify the Square Roots

To simplify each square root, factor the numbers inside the radicals into their prime factors.\[ \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \] \[ \sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2} \]

Step 2: Combine the Simplified Terms

Since both terms have the common radical \(\sqrt{2}\), you can add them directly: \[ 3\sqrt{2} + 4\sqrt{2} = (3 + 4)\sqrt{2} = 7\sqrt{2} \]

Step 3: Conclusion

The simplified form of \( \sqrt{18} + \sqrt{32} \) is \( 7\sqrt{2} \). There's no further simplification possible.

Key Concepts

Prime Factorization

Understanding prime factorization is crucial in the process of simplifying square roots. Essentially, any number can be expressed as a product of its prime factors. Prime factors are numbers greater than 1 that have no divisors other than 1 and themselves.
For example, to find the prime factorization of 18, we look for the smallest prime number that divides 18, which is 2. We divide 18 by 2 and get 9. Next, 9 is divided by 3 (since 3 is a prime number), resulting in 3. Finally, 3 itself is a prime number. Thus, the prime factorization of 18 is 2 x 3 x 3.
Similarly, for 32, the smallest prime factor is 2. Dividing 32 by 2 gives 16, again divide by 2 to get 8, and continue dividing by 2 until you get 1. The prime factorization of 32 is 2 x 2 x 2 x 2 x 2.
Understanding this concept helps us recognize which factors form perfect squares, which are essential to simplify square roots.

Combining Radicals

Combining radicals involves adding or subtracting terms that have a common radical part, much like combining like terms in algebra. This is only possible when the radicals are identical.

• For example, in the expression \(3\sqrt{2} + 4\sqrt{2}\), both terms have the common radical \(\sqrt{2}\).
• You can then combine these by adding the coefficients (the numbers in front of the radicals): 3 and 4.
• When added together, they form a single term \(7\sqrt{2}\).

This step is similar to factoring out the common radical term and simplifying the expression, making it easier to work with.

Square Roots Simplification

Simplifying square roots is a technique for expressing square roots in their simplest form, which often involves reducing the number under the radical sign to its smallest possible components.
We achieve this by using the prime factorization method discussed earlier. The goal is to identify perfect squares within the prime factors.

• For instance, \(\sqrt{18}\) can be split into \(\sqrt{9 \times 2}\).
• Since 9 is a perfect square (\(3^2\)), \(\sqrt{18}\) becomes \(3\sqrt{2}\).
• Similarly, \(\sqrt{32}\) can be expressed as \(\sqrt{16 \times 2}\), where 16 is a perfect square (\(4^2\)), resulting in \(4\sqrt{2}\).

Ultimately, simplifying square roots helps in reducing the problem to a more manageable form and enables further operations such as combining terms efficiently.