Problem 211 - Elementary Algebra

Question

In the following exercises, simplify. $$ \sqrt{175 k^{4}}-\sqrt{63 k^{4}} $$

Step-by-Step Solution

Answer

The simplified form is \(2k^2 \sqrt{7}\).

Step 1: Break down each square root

First, break down the numbers inside the square roots into their prime factors: \( \sqrt{175} = \sqrt{25 \times 7} = \sqrt{5^2 \times 7}\)Similarly, for 63: \( \sqrt{63} = \sqrt{9 \times 7} = \sqrt{3^2 \times 7} \)

Step 2: Apply the power property

Next, apply the property of square roots to separate the factors: \( \sqrt{175k^4} = \sqrt{(5^2 \times 7)k^4} = \sqrt{5^2}\sqrt{7}\sqrt{k^4}\)Using the same method for 63k^4: \( \sqrt{63k^4} = \sqrt{(3^2 \times 7)k^4} = \sqrt{3^2}\sqrt{7}\sqrt{k^4} \)

Step 3: Simplify each term

Simplify the square roots of the perfect squares and the exponent of k: \( \sqrt{5^2}\sqrt{k^4} = 5k^2\)So: \( \sqrt{175k^4} = 5k^2 \sqrt{7} \)Similarly, for 63k^4: \( \sqrt{3^2}\sqrt{k^4} = 3k^2 \)So: \( \sqrt{63k^4} = 3k^2 \sqrt{7} \)

Step 4: Subtract the simplified forms

Combine the simplified forms and subtract: \( 5k^2 \sqrt{7} - 3k^2 \sqrt{7} = (5k^2 - 3k^2) \sqrt{7} = 2k^2 \sqrt{7} \)

Key Concepts

Square Roots

A square root finds the number that, when multiplied by itself, equals the original number. For example, the square root of 9 is 3 because 3 multiplied by 3 is 9. You can represent square roots with the symbol \(\text{√}\). Understanding square roots is essential because it helps in breaking down and simplifying more complex expressions. In expressions like \(√{63k^4}\), focus on identifying which components are perfect squares. This step makes the simplification process easier by allowing you to separate these components efficiently.

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks, which are prime numbers. For example, the number 12 can be broken down into \(2 \times 2 \times 3\), where 2 and 3 are prime. To simplify radical expressions like \(√{175k^4}\) or \(√{63k^4}\), start by breaking down the numerical part into prime factors. For 175, the prime factorization is \(5^2 \times 7\). Similarly, for 63, it's \(3^2 \times 7\). This process makes it easier to identify and work with square root properties.

Exponent Properties

Exponents represent how many times a number is multiplied by itself. For instance, \(k^4\) means \(k \times k \times k \times k\). Important properties include the product of powers \(a^m \times a^n = a^{m+n}\) and the power of a power \( (a^m)^n = a^{mn} \). When simplifying expressions like \(√{(5^2 \times 7)k^4}\), take note of the exponents. The square root of \(k^4\) is \(k^2\), because \( (k^2)^2 = k^4 \). Recognizing these properties can streamline the simplification process.

Simplification

Simplifying an expression means making it as simple as possible without changing its value. In our example, after breaking down the numbers and applying square root properties, we simplify \(√{175k^4}\) to \(5k^2√7\). Similarly, \(√{63k^4}\) simplifies to \(3k^2√7\). The final step involves combining these simplified terms: \(5k^2√7 - 3k^2√7\). Because both terms share the \(√7\) component, we can directly subtract their coefficients: \(5k^2 - 3k^2 = 2k^2\). Thus, the simplified form is \(2k^2√7\).