Problem 65 - Algebra 2

Question

Simplify. $$ \sqrt{32}+\sqrt{18}-\sqrt{50} $$

Step-by-Step Solution

Answer

The simplified expression is \( 2\sqrt{2} \).

Step 1: Simplify each square root individually

To simplify the expression \( \sqrt{32} + \sqrt{18} - \sqrt{50} \), start by simplifying each square root separately. Break down each number into prime factors.\[\sqrt{32} = \sqrt{2^5} = \sqrt{2^4 \times 2} = 2^2 \sqrt{2} = 4\sqrt{2},\]\[\sqrt{18} = \sqrt{2 \times 3^2} = 3\sqrt{2},\]\[\sqrt{50} = \sqrt{2 \times 5^2} = 5\sqrt{2}.\]

Step 2: Substitute the simplified forms back into the expression

Now replace the square roots in the original expression with their simplified forms:\[4\sqrt{2} + 3\sqrt{2} - 5\sqrt{2}\]

Step 3: Combine like terms

Since all terms involve \( \sqrt{2} \), you can combine them:\[4\sqrt{2} + 3\sqrt{2} - 5\sqrt{2} = (4 + 3 - 5)\sqrt{2} = 2\sqrt{2}.\]

Key Concepts

Prime Factorization

To simplify a radical expression, prime factorization is a key starting point. This involves breaking down a number into its basic prime numbers -- the building blocks of all integers. For example, consider the expression \( \sqrt{32} + \sqrt{18} - \sqrt{50} \). Each number inside the square root needs to be expressed as a product of prime factors:

• For 32, which is 2 raised to the power of 5: \( 32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5 \).
• For 18, break it down as 2 times 3 squared: \( 18 = 2 \times 3 \times 3 = 2 \times 3^2 \).
• For 50, you have 2 times 5 squared: \( 50 = 2 \times 5^2 \).

Using prime factorization simplifies the process of dealing with square roots, which we'll explore more in the next section. The aim is to express each number in such a way that allows easy simplification with square roots.

Square Roots

The next task with radicals is simplifying them by using their prime factored forms. When dealing with square roots, we look for pairs of identical numbers because the square root of a product of a pair, say \( x^2 \), is simply \( x \). This is how we apply prime factorization to simplify square roots:

• For \( \sqrt{32} \): Recognize \( 2^5 \) as \( 2^4 \times 2 \). The \( 2^4 \) can be taken out of the square root as \( 2^2 = 4 \), leaving us with \( 4\sqrt{2} \).
• For \( \sqrt{18} \): You have \( 3^2 \times 2 \) inside the root. The \( 3^2 \) comes out as 3, resulting in \( 3\sqrt{2} \).
• For \( \sqrt{50} \): The \( 5^2 \) can be taken out as 5, simplifying to \( 5\sqrt{2} \).

Simplifying square roots in this way lets you work with each term more easily and sets us up perfectly to combine like terms.

Combining Like Terms

Once each radical has been simplified, we move on to combining like terms, which is a straightforward process. You only need to gather terms that have the same radical part. This means terms that have the same expression under the square root can be combined:

• In our case, the terms \( 4\sqrt{2} \), \( 3\sqrt{2} \), and \( 5\sqrt{2} \) all have the same radical, \( \sqrt{2} \).
• Add and subtract the coefficients, which are the numbers in front of the square root. So, you perform \( 4 + 3 - 5 = 2 \).

Thus, the expression simplifies to \( 2\sqrt{2} \). By combining like terms, we save ourselves from handling multiple terms, and the expression becomes much neater.