Problem 85 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ \sqrt{300 x y} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 10 \sqrt{3xy} \).

Step 1: Factor the Inside of the Square Root

First, let's factor 300 into its prime factors. 300 can be broken down as: \[ 300 = 2 imes 2 imes 3 imes 5 imes 5 \] Now the expression under the square root becomes:\[ \sqrt{2^2 \times 3 \times 5^2 \times x \times y} \]

Step 2: Separate Square and Non-Square Factors

Identify and separate the perfect squares from under the square root:\[ \sqrt{2^2} = 2 \, \text{and} \, \sqrt{5^2} = 5 \]We can take these perfect square roots outside the square root:\[ \sqrt{2^2 \times 5^2} = 2 \times 5 = 10 \]

Step 3: Simplify the Square Root Expression

Now express the original square root with the separated parts:\[ \sqrt{300xy} = \sqrt{2^2 \times 3 \times 5^2 \times x \times y} = 10 \times \sqrt{3xy} \]

Step 4: Combine and Write the Final Simplified Expression

Combine what we took out of the square root with what remains inside to write the final simplified expression:\[ 10 \sqrt{3xy} \]

Key Concepts

Understanding Prime Factorization

Prime factorization is a fundamental concept in simplifying square roots and is used to break down numbers into their basic building blocks, the prime numbers. Prime numbers are numbers that are only divisible by 1 and themselves, such as 2, 3, 5, 7, etc.

To factor a number into its prime components, you continuously divide it by the smallest prime number possible until all resulting factors are prime. For instance:

• Start with the number 300, since it is even, the smallest prime you can divide by is 2.
• Divide 300 by 2 to get 150.
• Repeat the division by 2: 150 divided by 2 gives you 75.
• Next, 75 is not divisible by 2, so you try the next smallest prime, which is 3, leaving you with 25.
• Dividing 25 by 5 (the next smallest prime number) twice, since 25 equals 5 times 5, concludes the factorization.

Eventually, you end up with the prime factors: 2, 2, 3, 5, and 5. This decomposition into primes is incredibly helpful for operations such as simplifying square roots, as you've seen in the step-by-step solution.

Grasping Perfect Squares

Perfect squares are numbers that can be expressed as the square of an integer. When simplifying square roots, identifying perfect squares is crucial because they simplify perfectly, allowing you to "remove" them from the square root.

In the square root simplification process, you look for multiples like 2², 3², 5² etc. Come back to the definition: a perfect square arises from multiplying a number by itself:

• The number 1, since 1 × 1 = 1.
• The number 4, since 2 × 2 = 4.
• The number 9, since 3 × 3 = 9.
• The number 16, since 4 × 4 = 16.

For instance, in the given exercise, 2² and 5² are perfect squares within the number 300. These form perfect squares because they are the squares of 2 and 5 respectively, which makes extraction simpler. Recognizing these patterns accelerates simplification and makes the process straightforward, ensuring you can effectively handle the expression under the square root.

Simplifying Algebraic Expressions

Algebraic expressions can be intimidating, especially when they involve variables under a square root. However, they can be simplified with a systematic approach, especially when combined with techniques like identifying perfect squares and prime factorization.

In expressions such as \ \sqrt{300xy}, variables like x and y are part of the mathematical expression and can be handled similarly to numbers:

• Follow the same prime factorization approach with numerical coefficients, but treat variables as separate entities.
• Identify if the variables themselves are part of squares that can be simplified further. For instance, if \(x\) or \(y\) is given as \(x^2\) or \(y^2\), they can be treated as perfect squares.
• Combine the simplified numerical part and the variable part to form the final expression.

So, while x and y here don't form perfect squares as isolated variables, they contribute to the complete expression under the square root. In our example, after treating 300 with prime factorization and correlating with perfect squares, x and y remain under the square root as they don't form a perfect square, leading to the final simplified form of 10 \( \sqrt{3xy} \). Always remember, practice makes perfect—breaking down each component is key to mastering algebraic simplifications!