Problem 19 - Prealgebra

Question

Simplify each square root, then combine if possible. Assume all variables represent positive numbers. $$\sqrt{72 x^{2}}-\sqrt{50 x^{2}}$$

Step-by-Step Solution

Answer

\( x\sqrt{2} \)

Step 1: Simplify \( \sqrt{72x^2} \)

First, factor the number 72 into its prime factors: \( 72 = 2^3 \cdot 3^2 \). So, \( \sqrt{72x^2} = \sqrt{2^3 \cdot 3^2 \cdot x^2} \). This can be broken down into: \( \sqrt{2^3} \cdot \sqrt{3^2} \cdot \sqrt{x^2} \). Simplifying these we get \( \sqrt{2^3} = \sqrt{8} = 2 \sqrt{2} \), \( \sqrt{3^2} = 3 \), and \( \sqrt{x^2} = x \). Therefore, \( \sqrt{72x^2} = 6x\sqrt{2} \).

Step 2: Simplify \( \sqrt{50x^2} \)

Next, factor the number 50 into its prime factors: \( 50 = 2 \cdot 5^2 \). So, \( \sqrt{50x^2} = \sqrt{2 \cdot 5^2 \cdot x^2} \). This can be broken down into: \( \sqrt{2} \cdot \sqrt{5^2} \cdot \sqrt{x^2} \). Simplifying these we get \( \sqrt{5^2} = 5 \) and \( \sqrt{x^2} = x \). Therefore, \( \sqrt{50x^2} = 5x\sqrt{2} \).

Step 3: Subtract the Simplified Expressions

Now we have the simplified expressions: \( 6x\sqrt{2} \) and \( 5x\sqrt{2} \). To combine them, simply subtract the coefficients: \( 6x\sqrt{2} - 5x\sqrt{2} = (6x - 5x)\sqrt{2} = x\sqrt{2} \).

Key Concepts

Square Root Properties

Square roots are fascinating because they allow us to understand and work with numbers in an entirely different way. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because \(3 \times 3 = 9\).

To simplify square roots, especially when dealing with variables like in \( \sqrt{72x^2} \), understanding the properties of square roots is crucial. Square root properties include:

• \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \): You can break down the square root of a product into the product of their square roots.
• \( \sqrt{a^2} = a \): If \(a\) is a non-negative number, the square root of \(a^2\) is simply \(a\).

These properties are particularly helpful for simplifying algebraic expressions under the square root.

Prime Factorization

Prime factorization is a method used to break down a number into its basic building blocks - the prime numbers. This is extremely useful when simplifying square roots.

Let's take the number 72 as an example: To factor 72, you divide it by the smallest prime number possible. Start with 2:

• 72 divided by 2 is 36.
• 36 divided by 2 is 18.
• 18 divided by 2 is 9.
• Finally, 9 divided by 3 is 3, and then 3 divided by 3 is 1.

The prime factors of 72 are therefore \(2^3\) and \(3^2\). Using these factors, we can then simplify square roots by pairing identical factors under the radical, as seen in \(\sqrt{72} = \sqrt{2^3 \times 3^2} = 6\sqrt{2}\). The same process is used for other numbers like 50 in \(\sqrt{50} = \sqrt{2 \times 5^2} = 5\sqrt{2}\).

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operational signs. In our exercise, we are working with expressions that include the variable \(x\).

When simplifying square roots with variables, it's crucial to treat the variable part separately, just like we do with the numbers. For example, in the expression \(\sqrt{72x^2}\), after simplifying the numerical part using prime factorization, we focus on the variable. Since \(x^2\) is a perfect square, its square root is simply \(x\). This simplification process helps us handle larger and more complex expressions with ease. It's essential to assume all variables represent positive numbers in these cases, ensuring a positive square root outcome for \(x\).

Coefficient Subtraction

Coefficient subtraction is a technique used when simplifying and combining expressions that have like terms. Coefficients are the numerical parts of algebraic expressions. When you have terms that involve the same radicals, such as \(6x\sqrt{2}\) and \(5x\sqrt{2}\), you can combine them by subtracting their coefficients.

Here's how it works: first align your like terms, in this case, expressions that share the common \(\sqrt{2}\). Then, you simply perform subtraction on their coefficients:

• \(6x\sqrt{2} - 5x\sqrt{2}\) becomes \((6x - 5x)\sqrt{2}\)
• Which simplifies to \(x\sqrt{2}\)

Combining like terms through coefficient subtraction is a valuable skill in algebra, as it simplifies complex expressions and makes calculations much more manageable.