Problem 30 - Precalculus Mathematics for Calculus

Question

Simplify the expression. $$\sqrt{75}+\sqrt{48}$$

Step-by-Step Solution

Answer

The simplified expression is \(9\sqrt{3}\).

Step 1: Break Down the Radicals into Prime Factors

To simplify the expression, start by breaking down each number under the square root into its prime factors. - For 75: The prime factors are 3 and 5 (since 75 = 3 x 5 x 5). - For 48: The prime factors are 2 and 3 (since 48 = 2 x 2 x 2 x 2 x 3).

Step 2: Simplify Each Square Root

Next, simplify each square root by taking out pairs of numbers from under the radical sign:- For \( \sqrt{75} \): - Since 75 = 3 x 5 x 5, you can take out a pair of 5. Thus, \( \sqrt{75} = 5\sqrt{3} \).- For \( \sqrt{48} \): - Since 48 = 2 x 2 x 2 x 2 x 3, you can take out two pairs of 2. Thus, \( \sqrt{48} = 4\sqrt{3} \).

Step 3: Add the Simplified Radicals

After simplifying the radicals, you end up with two terms that contain the same radical, \( \sqrt{3} \). So, you can add them together:- \( 5\sqrt{3} + 4\sqrt{3} = (5 + 4)\sqrt{3} = 9\sqrt{3} \).

Key Concepts

Prime Factorization

When we simplify radical expressions, prime factorization is an important first step. Prime factorization involves breaking down a number into its smallest components, which are prime numbers. A prime number is a number greater than 1 that cannot be formed by multiplying two smaller numbers. For our example, we find the prime factors of 75 and 48.

• For 75: 75 can be broken down into 5 x 5 x 3, giving the prime factorization of 3 and 5.
• For 48: 48 is decomposed into 2 x 2 x 2 x 2 x 3, yielding the prime factors 2 and 3.

Understanding prime factorization helps us simplify radicals by identifying pairs of identical numbers, which can be extracted to simplify the square root.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and so forth. A square root is the most common, noted as \( \sqrt{} \), and it answers the question, 'What number multiplied by itself gives this value?'. In our case, we deal with square roots in the expression \( \sqrt{75} + \sqrt{48} \).

To simplify these expressions, we first use the prime factorization method. We observe the prime factor components of 75 and 48 to find perfect squares within each radical. When removing these squares, you reduce the complexity of the expression. This means taking out pairs from under the radical sign:

- For \( \sqrt{75} \), the pair 5 is squared inside, so you pull it out, simplifying to \( 5\sqrt{3} \).- For \( \sqrt{48} \), the two pairs of 2 are squared inside, extracting them simplifies it to \( 4\sqrt{3} \).

Algebraic Simplification

Algebraic simplification of radicals involves combining like terms, once they have been simplified. In \( \sqrt{75} + \sqrt{48} \), each radical was simplified to \( 5\sqrt{3} \) and \( 4\sqrt{3} \) respectively. Both terms now have the same radical base, \( \sqrt{3} \).

To simplify further, you add the coefficients (the numbers outside the radical) together while keeping the radical part unchanged:

• Combine the coefficients: \( 5 + 4 \)
• This equals: \( 9 \)

Thus, the expression \( 5\sqrt{3} + 4\sqrt{3} \) simplifies to \( 9\sqrt{3} \). This represents the cleanest and simplest form of the original radical expression. Understanding these steps ensures precision in handling similar algebraic tasks.