Problem 185 - Elementary Algebra

Question

In the following exercises, simplify. $$ 2 \sqrt{50}-3 \sqrt{72} $$

Step-by-Step Solution

Answer

-8 \sqrt{2}

Step 1: Break Down Each Radical

First, simplify each square root separately. Find the prime factorization of the numbers inside the radicals: \(50 = 2 \times 5^2\) and \(72 = 2^3 \times 3^2\).

Step 2: Simplify Each Square Root

Simplify the radical by taking out the squares: \(\sqrt{50} = \sqrt{2 \times 5^2} = 5\sqrt{2}\) and \(\sqrt{72} = \sqrt{2^3 \times 3^2} = 6\sqrt{2}\).

Step 3: Multiply By the Coefficients

Multiply the simplified square roots by their coefficients: \(2 \sqrt{50} = 2 \times 5 \sqrt{2} = 10\sqrt{2}\) and \(-3 \sqrt{72} = -3 \times 6 \sqrt{2} = -18\sqrt{2}\).

Step 4: Combine Like Terms

Combine the like terms: \(10\sqrt{2} - 18\sqrt{2} = (10 - 18)\sqrt{2} = -8\sqrt{2}\).

Key Concepts

Prime Factorization

Prime factorization is a fundamental concept in mathematics used to break down numbers into their prime components. Prime numbers are numbers that are only divisible by 1 and themselves. Examples include 2, 3, 5, and 7. To find the prime factorization of a number, you repeatedly divide by the smallest prime number until the resulting quotient is a prime number.

For instance, let's take the number 50:
- First, divide 50 by 2 (the smallest prime number), which gives 25.
- Now, divide 25 by 5, another prime number, giving 5.
- Lastly, 5 is already a prime number. So the prime factorization of 50 is \( 2 \times 5^2 \).

Using prime factorization helps to simplify square roots, as shown in our exercise. Similarly, for 72:
- Divide 72 by 2 repeatedly until you no longer can. This gives you \( 2^3 \).
- Then, divide by 3 until you can't anymore, resulting in \( 3^2 \).
- Thus, the prime factorization of 72 is \( 2^3 \times 3^2 \). Prime factorization helps simplify complex expressions by breaking them down to their base elements.

Square Roots

Simplifying square roots involves finding the principal square root of a number, which is the non-negative root. The square root of \( n \) is a value that, when multiplied by itself, gives \( n \). Recognizing how the prime factorization aids in this process is vital.

Take the square root of 50 from the exercise. Using prime factorization:
- \( \text{50} = 2 \times 5^2 \)
- The square root of 5 squared becomes 5, because \( \text{(5}^{2}) = 5 \times 5 \). So, \( \text{√50} \) simplifies to \( \text{5√2} \).

Similarly, for 72:
- \( \text{72} = 2^3 \times 3^2 \)
- Interpreting the prime factors in pairs: \( \text{(2×2)} = 4 \) and \( \text{(3}^{2}) = 9 \). The pair factors become: \( 2\sqrt{2} \) meaning remain there, hence, \( 6√2 \).

This results in the simplified form of the square root of the numbers, making further calculations easier.

Combining Like Terms

Combining like terms is an essential algebraic technique for simplifying expressions. Like terms are terms whose variables (and their exponents) are the same. This concept extends to radicals when the radicals have the same radicand.

Within our exercise, combining like terms refers to terms involving \( \text{√2} \):
- After simplifying \( \text{2√50} \) and \( \text{-3√72} \), we get \( \text{10√2} \) and \( -18√2 \).
- As they both contain \( \text{√2} \), they are like terms and can be combined directly.

The next step is to add or subtract the coefficients:
- Here, it ends up being \( 10 \text{√2} - 18 \text{√2} \).
- Subtract coefficients: \( 10 - 18 = -8 \).

Hence, the combined term is \( -8 \text{√2} \). This approach helps in cleansing and harmonization to obtain the final simplified form.