Problem 23 - College Algebra

Question

Simplify each expression. $$(\sqrt{42})(\sqrt{30})$$

Step-by-Step Solution

Answer

The expression simplifies to \(6\sqrt{35}\).

Step 1: Use the Property of Square Roots

The property of square roots states that the product of two square roots is the square root of the product of the numbers. Therefore, we can start by writing \((\sqrt{42})(\sqrt{30})\) as \(\sqrt{42 \times 30}\).

Step 2: Calculate the Product under the Square Root

Multiply the numbers under the square root: \(42 \times 30 = 1260\). Thus, \(\sqrt{42 \times 30} = \sqrt{1260}\).

Step 3: Simplify the Square Root

Next, simplify \(\sqrt{1260}\). Begin by factorizing 1260 into its prime factors: \(1260 = 2^2 \times 3^2 \times 5 \times 7\).

Step 4: Apply the Square Root to the Prime Factors

Apply the square root to the prime factorization: \(\sqrt{1260} = \sqrt{2^2 \times 3^2 \times 5 \times 7}\). Separate the perfect squares: \(\sqrt{2^2} \times \sqrt{3^2} \times \sqrt{5} \times \sqrt{7} = 2 \times 3 \times \sqrt{5 \times 7}\).

Step 5: Multiply and Simplify

Multiply the integers: \(2 \times 3 = 6\). Thus, \(6 \times \sqrt{35} = 6\sqrt{35}\). The expression \(\sqrt{42} \times \sqrt{30}\) simplifies to \(6\sqrt{35}\).

Key Concepts

Property of Square Roots

One of the fundamental properties of square roots is the way they handle the multiplication of numbers. This property states:

• The square root of a product is the product of the square roots of the individual factors.

This means for any two numbers, say, \(a\) and \(b\), the expression \(\sqrt{a} \times \sqrt{b}\) is equal to \(\sqrt{a \times b}\). This property is incredibly useful because it allows us to combine square roots, making expressions simpler or easier to work with.
By using this property, we transformed \((\sqrt{42})(\sqrt{30})\) into \(\sqrt{42 \times 30}\). This is an essential step in simplifying expressions as it consolidates multiple square roots into a single expression.
It's important to remember that this property only holds true for non-negative numbers, as square roots of negative numbers delve into complex number territory.

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks, called prime numbers.

• A prime number is one that can only be divided by 1 and itself without leaving a remainder.

Understanding prime factorization helps in various mathematical situations, especially when dealing with square roots, as it makes simplification much easier.
To factor the number 1260, we find its prime factors step by step:

• Divide by 2: \(1260 \div 2 = 630\) (repeat once more since 630 is also divisible by 2)
• Divide by 3: \(315 \div 3 = 105\) (repeat once more with the result)
• Divide by 5: \(35 \div 5 = 7\)
• Finally, 7 is already a prime number.

Putting it together, \(1260 = 2^2 \times 3^2 \times 5 \times 7\). Leveraging this knowledge, we can now more effectively simplify square roots.
This method is efficient and systematic, ensuring none of the prime factors are overlooked.

Square Root Simplification

Simplifying square roots involves reducing the expression to its simplest form by removing perfect squares from under the square root.
Using the prime factorization we found earlier, \(\sqrt{1260} = \sqrt{2^2 \times 3^2 \times 5 \times 7}\), we can separate the perfect squares.

• Extract each perfect square from the square root: \(\sqrt{2^2} = 2\) and \(\sqrt{3^2} = 3\).

This adjusted the expression to: \(2 \times 3 \times \sqrt{5 \times 7}\).
We multiply the whole numbers outside the square root, giving us \(6 \times \sqrt{35}\). The expression is fully simplified.
This method is very helpful as it provides a clearer and more concise form. While looking complex initially, using prime factorization followed by simplification transforms challenging expressions into manageable forms, firmly grounding understanding of square roots and algebraic simplifications.