Problem 56 - Intermediate Algebra

Question

Simplify. See Examples 3 and 4 $$ \sqrt{12 r^{9} s^{12}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 2r^4s^6\sqrt{3r} \).

Step 1: Prime Factorization of 12

To simplify the expression, we start by simplifying the square root of 12. First, we express 12 as a product of its prime factors: \( 12 = 2^2 \times 3 \). This will help us find the square root.

Step 2: Split the Radicand

Next, split the original expression inside the square root as separate components: \( \sqrt{12 r^9 s^{12}} = \sqrt{2^2 \times 3 \times r^9 \times s^{12}} \). Now, consider each factor separately.

Step 3: Simplify Square Root of Numbers

Take the square root of the numerical part inside the square root: \( \sqrt{2^2 \times 3} = 2\sqrt{3} \), because the square root of \( 2^2 \) is 2, and 3 stays under the square root.

Step 4: Simplify Variables with Even Powers

Apply the square root to \( s^{12} \): Since the exponent is even, \( \sqrt{s^{12}} = s^6 \). This is because \( s^{12} = (s^6)^2 \), and the square root of \( (s^6)^2 \) is \( s^6 \).

Step 5: Simplify Variables with Odd Powers

Deal with \( r^9 \): Since 9 is odd, express \( r^9 = r^8 \times r \) where \( r^8 = (r^4)^2 \). The square root of \( r^8 \) is \( r^4 \), thus, \( \sqrt{r^9} = r^4 \sqrt{r} \).

Step 6: Combine All Simplified Components

Combine all parts together: \( \sqrt{12 r^9 s^{12}} = 2 r^4 s^6 \sqrt{3r} \). Therefore, the expression is simplified successfully.

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into its prime number components. A prime number is a number greater than one that has no divisors besides one and itself.
To simplify a square root, like the square root of 12, we first find the prime factorization of 12.

• Start by dividing 12 by the smallest prime number, which is 2. We get 12 = 2 × 6.
• Keep dividing by 2 until you can no longer divide evenly. So, 6 can be divided by 2 to get 3. Now, 12 is expressed as 2 × 2 × 3.
• This translates to 12 = 2² × 3 in terms of prime factors.

Understanding prime factorization is crucial, as it aids in simplifying square roots by grouping pairs of prime factors.

Square Root Properties

Understanding the properties of square roots is essential when simplifying expressions. A square root surrounds a number or a variable, and the goal is to simplify it by extracting perfect squares.
Here are some basic properties:

• The square root of a product can be split into the product of square roots: \( \sqrt{ab} = \sqrt{a}\sqrt{b} \).
• The square root of a square is the original number: \( \sqrt{a^2} = a\).
• If a factor inside a square root is a perfect square, you can take it out of the square root.

For example, \( \sqrt{2^2 \times 3} = 2\sqrt{3} \) because \( \sqrt{2^2} = 2 \), simplifying just the perfect squares helps reduce the expression.

Simplifying Variable Expressions

When simplifying expressions that include variables under a square root, it's important to consider the exponents of the variables.
Here's how you simplify based on the exponent's parity:

• If the exponent is even, you can fully take that variable out of the square root. For example, \( \sqrt{s^{12}} = s^6 \) since \( s^{12} = (s^6)^2 \).
• If the exponent is odd, separate it into an even part and a leftover 1. For example, \( r^9 = r^8 \times r = (r^4)^2 \times r \), and hence \( \sqrt{r^9} = r^4\sqrt{r} \).

This approach encourages focusing on getting as many complete squares as possible out of the square root, thus further simplifying the expression.

Mathematical Expressions

A mathematical expression consists of numbers, variables, and different operations. Understanding how to simplify such expressions is key for solving problems efficiently.
Mathematical expressions often involve:

• Numbers, as coefficients or standalone integers.
• Variables, which stand for unspecified numbers.
• Operations like addition, subtraction, multiplication, or division.

When simplifying expressions, it is about combining like terms, reducing complex expressions, and making calculations easier. Whether dealing with coefficients, variables, or radicals, the understanding and application of simplification rules are what make solving mathematical problems more straightforward.