Problem 30 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ 6 \sqrt{18}+5 \sqrt{32}+4 \sqrt{50} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(6 \sqrt{18} + 5 \sqrt{32} + 4 \sqrt{50}\). Answer: The simplified expression is \(58 \sqrt{2}\).

Step 1: Break down each term inside the square root

We will find the prime factors of each number inside the square root. \(18 = 2 \times 3^2\) (prime factors: 2, 3) \(32 = 2^5\) (prime factors: 2) \(50 = 2 \times 5^2\) (prime factors: 2, 5) Now we can rewrite the expression with the prime factors. $$ 6 \sqrt{2 \times 3^2} + 5 \sqrt{2^5} + 4 \sqrt{2 \times 5^2} $$

Step 2: Simplify the square roots

We can now simplify the square roots by taking out the factor which has an even power. $$ 6 \times 3 \sqrt{2} + 5 \times 2^2 \sqrt{2} + 4 \times 5 \sqrt{2} $$

Step 3: Simplify the constants outside the square roots

We can now simplify the constants outside the square root. $$ 18 \sqrt{2} + 20 \sqrt{2} + 20 \sqrt{2} $$

Step 4: Combine like terms

Since all terms are like terms (with a common square root), we can combine them. $$ (18 + 20 + 20) \sqrt{2} $$ $$ 58 \sqrt{2} $$ The simplified expression is \(58 \sqrt{2}\).

Key Concepts

Radical Expressions

Understanding radical expressions is an essential step in simplifying square roots. A radical expression contains a radical symbol () with a number or expression underneath, known as the radicand. In expressions like ), the goal is to simplify the number under the radical to its simplest form. Simplification often involves identifying and pulling out perfect squares from under the radical, because the square root of a perfect square is an integer. This process makes it easier to perform further operations, such as addition or multiplication, with other square roots. In our exercise, simplification involved breaking down the radicands (18, 32, and 50) to their prime factors and then extracting factors that are perfect squares.

Prime Factorization

Prime factorization is the process of expressing a number as the product of its prime factors. This is a key step in simplifying square roots as it allows for the identification of squares within the radicand which can be simplified outside the radical sign.

For instance, the number 18 can be factored into prime numbers as follows: - Factoring 18 gives us 2 and 9. - 9 can further be broken down into 3 times 3, which are prime numbers. So 18 can be written as 2 times 3 squared (2 × 3²). Recognizing that 3² is a perfect square, we can pull 3 out of the square root. As seen in the exercise, breaking down radicands to their prime factors enables us to simplify each square root before combining like terms.

Combining Like Terms

Combining like terms is a process often employed in algebra to simplify expressions. Terms are 'like' if they have the same variable parts or, in the case of square roots, the same radicand. Once the radical expressions are simplified and in their simplest form, we can then add or subtract the coefficients, just as done with regular numbers.

Following the simplification of the individual square roots in the exercise, we grouped like terms together. Because each term had a ) factor, we were able to combine the numerical coefficients (in this case, 18, 20, and 20) to achieve a final simplified form of ). This step is vital for reaching a concise expression, allowing for easier interpretation and further manipulation if necessary.