Problem 28 - Algebra and Trigonometry

Question

Simplify the expression. \(\sqrt{8}+\sqrt{50}\)

Step-by-Step Solution

Answer

The simplified expression is \(7\sqrt{2}\).

Step 1: Prime Factorization of 8

First, break down 8 into its prime factors: \[8 = 2 \times 2 \times 2 = 2^3\].

Step 2: Simplify \(\sqrt{8}\)

Use the prime factorization to simplify \(\sqrt{8}\):\[\sqrt{8} = \sqrt{2^3} = \sqrt{2^2 \times 2} = \sqrt{(2^2) \times 2} = 2\sqrt{2}\].

Step 3: Prime Factorization of 50

Next, break down 50 into its prime factors:\[50 = 2 \times 5 \times 5 = 2 \times 5^2\].

Step 4: Simplify \(\sqrt{50}\)

Use the prime factorization to simplify \(\sqrt{50}\):\[\sqrt{50} = \sqrt{2 \times 5^2} = \sqrt{2 \times (5^2)} = 5\sqrt{2}\].

Step 5: Combine Like Terms

Combine the simplified radicals:\[\sqrt{8} + \sqrt{50} = 2\sqrt{2} + 5\sqrt{2}\].As both terms have the same radical part \(\sqrt{2}\), you can add the coefficients together:\[2\sqrt{2} + 5\sqrt{2} = (2+5)\sqrt{2} = 7\sqrt{2}\].

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its simplest building blocks, known as prime numbers. It is a key step used in simplifying radical expressions. To factor a number, we repeatedly divide it by the smallest possible prime number until only prime numbers remain.
For the number 8, we start by dividing by 2 because it is the smallest prime number:

• 8 divided by 2 equals 4, so we write this as 8 = 2 × 4.


• Next, 4 is divided by 2 again to give 2, so we have 8 = 2 × 2 × 2, which we can write as 8 = 2^3.

Now, let's factor 50, following the same process:

• 50 divided by 2 is 25, so we start with 50 = 2 × 25.


• Then 25 is divided by 5 giving us 5, so we have 50 = 2 × 5 × 5, or more neatly, 50 = 2 × 5^2.

Prime factorization simplifies our numbers into a useful form that we can work with when simplifying radicals.

Radical Simplification

Radical simplification involves taking a complex square root expression and breaking it down into a simpler form. This is done using the prime factors of the number inside the square root.
To simplify \( \sqrt{8} \), we use its prime factorization:

• Start by expressing \( \sqrt{8} \) as \( \sqrt{2^3} \).


• Rewrite that as \( \sqrt{2^2 \times 2} \). Using the property of radicals that states \( \sqrt{a^2} = a \), we take \( 2^2 \) outside of the square root: \( \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \).

Next, let's simplify \( \sqrt{50} \) using the same technique:

• The factorization of 50 is already found, written as \( \sqrt{2 \times 5^2} \).


• Applying the same radical property, \( \sqrt{5^2} \) becomes 5, simplifying it to \( 5\sqrt{2} \).

By reducing each square root to its simplest form, simplification makes it easier to add radical expressions.

Adding Radical Expressions

Adding radical expressions is like combining like terms in algebra. To do this, the expressions must have the same radical part. If they do, you can simply add the numbers in front of the radicals, known as the coefficients.
For the expression \( \sqrt{8} + \sqrt{50} \), we've simplified \( \sqrt{8} \) to \( 2\sqrt{2} \) and \( \sqrt{50} \) to \( 5\sqrt{2} \). Both have the same radical component (\( \sqrt{2} \)), meaning we can combine them:

• Write the expression as \( 2\sqrt{2} + 5\sqrt{2} \).


• The coefficients \( 2 \) and \( 5 \) can be added together: \( 2 + 5 = 7 \).


• Finally, the result is \( 7\sqrt{2} \).

Ensure the radicals are similar before adding; otherwise, they cannot be combined this way. This simplifies complex expressions into more manageable forms.