Problem 23 - Algebra 1

Question

Simplify the expression. $$\sqrt{75}+\sqrt{3}$$

Step-by-Step Solution

Answer

\(\sqrt{75}+\sqrt{3}\) simplifies to \(6\sqrt{3}\).

Step 1: Simplify \(\sqrt{75}\)

Find the prime factors of 75. The result is \(5 \times 5 \times 3\). Alternatively, it can also be expressed as \(5^2 \times 3\). The perfect square here is \(5^2\), meaning we can take out this square root. \(\sqrt{75}\) simplifies to \(5\sqrt{3}\).

Step 2: Rewrite the Expression

Replace \(\sqrt{75}\) in the original expression with \(5\sqrt{3}\), resulting in \(5\sqrt{3} + \sqrt{3}\).

Step 3: Simplify the Expression

The above expression can be simplified by combining like terms. The root values are the same, so the coefficients can be added together to give \(6\sqrt{3}\).

Key Concepts

Prime Factorization

When simplifying radical expressions, prime factorization is a helpful technique used to break down numbers into their simplest form. Prime factorization means expressing a number as a product of its prime numbers. Prime numbers are those which are only divisible by 1 and themselves.

For example, consider the number 75. To find its prime factors, start by dividing by the smallest prime number, which is 2. Since 75 is an odd number, it isn't divisible by 2. Move to the next prime number, which is 3. Divide 75 by 3, resulting in 25, as 75 divided by 3 gives 25.

Next, divide 25 by the smallest prime number it is divisible by, which is 5. So, 25 divided by 5 is 5. Finally, take 5 divided by itself, which gives 1, indicating that we have found all the prime factors. Therefore, the prime factorization of 75 is \( 3 \times 5 \times 5 \) or \( 3 \times 5^2 \). This expression helps in identifying any perfect squares within the radical.

Perfect Squares

Perfect squares play a critical role when simplifying square roots. A perfect square is a number which can be expressed as the square of an integer. Common examples are 4, 9, 16, and 25, as they are squares of 2, 3, 4, and 5 respectively.

In our exercise, by using prime factorization of 75, namely \( 5^2 \times 3 \), we identify \( 5^2 \) as a perfect square. The number 5 squared is 25, which is a perfect square. Thus, \( \sqrt{5^2} \) equates to 5.

When simplifying \( \sqrt{75} \), the perfect square \( 5^2 \) allows us to take 5 out of the square root, resulting in \( 5\sqrt{3} \). Identifying and extracting perfect squares allows us to make radical expressions simpler.

Combining Like Terms

Combining like terms is essential for simplifying expressions, including those with radicals. The concept involves adding or subtracting terms with the same variable part. For example, \( x + 2x \) equals \( 3x \) because both terms contain the variable \( x \).

In the context of our exercise, \( 5\sqrt{3} + \sqrt{3} \) contains like terms involving \( \sqrt{3} \). Think of \( \sqrt{3} \) as a common factor, similar to a variable. \( 5\sqrt{3} \) means "5 times \( \sqrt{3} \)", while \( \sqrt{3} \) means "1 times \( \sqrt{3} \)".

To combine these, add the coefficients (5 and 1) together, retaining the \( \sqrt{3} \), resulting in \( 6\sqrt{3} \). This simplification streamlines the expression by consolidating similar terms.