Problem 213 - Elementary Algebra

Question

In the following exercises, simplify. $$ 2 \sqrt{363}-2 \sqrt{300} $$

Step-by-Step Solution

Answer

2 \sqrt{3}

Step 1: Factorize inside the square roots

To simplify the given expression, first factorize the numbers inside the square roots. For 363: \[ 363 = 3 \times 121 = 3 \times 11^2 \]For 300: \[ 300 = 3 \times 100 = 3 \times 10^2 \]

Step 2: Simplify the square roots

Express the square roots using their factors: \[ 2 \times \sqrt{363} = 2 \times \sqrt{3 \times 11^2} \]\[ 2 \times \sqrt{300} = 2 \times \sqrt{3 \times 10^2} \]Since \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \), we can rewrite: \[ 2 \times \sqrt{3 \times 11^2} = 2 \times \sqrt{3} \times 11 \]\[ 2 \times \sqrt{3 \times 10^2} = 2 \times \sqrt{3} \times 10 \]

Step 3: Multiply and combine like terms

Multiply the constants and simplify: \[ 2 \times 11 \times \sqrt{3} - 2 \times 10 \times \sqrt{3} = 22 \sqrt{3} - 20 \sqrt{3} \]Now, combine the like terms: \[ (22 - 20) \sqrt{3} = 2 \sqrt{3} \]

Key Concepts

Factorization

Factorization is the process of breaking down a number into its constituent prime factors. This is useful when simplifying expressions involving radicals.
Let's look at our specific example. We begin with two numbers, 363 and 300, and factorize them:

• For 363, note that it equals 3 times 121, and further, 121 is the square of 11. So 363 can be written as:
  • 363 = 3 × 121
  • = 3 × 11².
• For 300, note that it equals 3 times 100, and further, 100 is the square of 10. So 300 can be written as:
  • 300 = 3 × 100
  • = 3 × 10².

This completes the factorization step.

Simplification of Radicals

Simplification of radicals involves expressing the square root in a simpler form using the factors we found in the previous step. Here's how we do that:
When you have an expression like

• For 2 √363, we can use the factorized form of 363:
  2 √(3 × 11²) and apply the property of radicals which states √(a × b) = √a × √b. This gives us 2 √(3 × 11²) = 2 √3 × √11² = 2 √3 × 11 = 22 √3.
• Similarly, for 2 √300, we proceed by substituting its factors: 2 √(3 × 10²). Using the same property, we get 2 √(3 × 10²) = 2 √3 × √10² = 2 √3 × 10 = 20 √3.

Simply put, the radical terms are simplified to their most reduced forms, making the rest of the problem easier to handle.

Like Terms in Algebra

Combining like terms is an essential algebraic skill. Like terms are terms that contain the same variable raised to the same power. In our example, after simplifying the radicals, we end up with 22 √3 and 20 √3.

Because both terms contain √3, they are like terms and can be combined:

• 22√3 - 20√3

We treat √3 as a common factor, just like combining 22x and 20x in algebra:

• Subtract the coefficients: (22 - 20) √3 = 2 √3.

Remember that combining like terms simplifies the expression. Here it reduces down to a single term: 2 √3.

This step is vital in algebra as it makes expressions more manageable and easier to understand.