Problem 222 - Elementary Algebra

Question

In the following exercises, simplify. $$ \sqrt{500}+\sqrt{405} $$

Step-by-Step Solution

Answer

19\sqrt{5}

Step 1: Simplify each square root individually

Begin by expressing each square root as the product of prime factors. For \(\sqrt{500}\), write it as \(\sqrt{100 \times 5} = \sqrt{100} \times \sqrt{5}\), which simplifies to \(10\sqrt{5}\). For \(\sqrt{405}\), write it as \(\sqrt{81 \times 5} = \sqrt{81} \times \sqrt{5}\), which simplifies to \(9\sqrt{5}\).

Step 2: Add the simplified square roots

Now add the simplified forms: \(10\sqrt{5} + 9\sqrt{5}\). Since both terms have the same radical part \(\sqrt{5}\), they can be combined as like terms.

Step 3: Combine like terms

Combine the coefficients: \(10 + 9 = 19\). This results in: \(19\sqrt{5}\).

Key Concepts

Prime Factorization

Prime factorization is a key step when simplifying square roots. This method involves breaking down a number into its prime factors, which are prime numbers that can be multiplied together to get the original number.
For example, when simplifying \(\root 500\), begin by finding the prime factors:

• 500 is even, so divide by 2: 500 ÷ 2 = 250.
• 250 is even, so divide by 2 again: 250 ÷ 2 = 125.
• 125 ends in 5, so divide by 5: 125 ÷ 5 = 25.
• 25 ends in 5, so divide by 5: 25 ÷ 5 = 5.
• 5 is a prime number.

Thus, the prime factors of 500 are 2, 2, 5, 5, and 5. This allows us to rewrite \(\root 500\) as\(\root(2^(2) \times 5^(3))\). By taking pairs of prime factors out of the root, you simplify it to \(\root 100 \times \root 5 = 10 \root 5\).

Combining Like Terms

Combining like terms is a fundamental technique in algebra where terms with the same variables and powers are simplified together.
In the context of our problem, after simplifying \(\root 500\) and \(\root 405\) separately, we get \(\root 10 \root 5\) and \(\root 9 \root 5\), respectively. Notice that both results contain \(\root 5\). Terms that contain the same radical part—here, \(\root 5\)—can be combined.
Thus, \(\root 10 \root 5 + 9 \root 5\) is similar to combining \((10x + 9x)\) in regular algebra. We add the coefficients: \(\root10 + 9 = 19\). Therefore, we get \(\root 19 \root 5\).

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because \(\root 4 x 4 = 16\). Sometimes, square roots can be simplified by using the prime factorization technique.
The general approach to simplification involves:

• Finding the prime factors of the number.
• Grouping the prime factors into pairs.
• Taking one number out of each pair out of the square root.
• Multiplying the numbers taken out by any remaining factors inside the root.

In our exercise, we simplified \(\root 500\) to \(\root 10 \root 5\) and \(\root 405\) to \(\root 9 \root 5\). The end result involves understanding that if two terms have the same square root, they can be combined like algebraic terms.
Keep practicing this method to solidify the concept of simplifying square roots and combining like terms. It is a crucial skill in algebra.