Problem 14 - Elementary Algebra

Question

Find each of the following products. $$ \sqrt{45} \sqrt{50} $$

Step-by-Step Solution

Answer

Answer: The product of the square roots of 45 and 50 is $15\sqrt{10}$.

Step 1: Simplify the square roots

First, we will break down each square root into its prime factors: $$ \sqrt{45} = \sqrt{9\cdot5} = \sqrt{3^2 \cdot 5}\\ \sqrt{50} = \sqrt{25\cdot2} = \sqrt{5^2 \cdot 2} $$ Then, we can simplify each square root by taking the square root of each perfect square factors (if any): $$ \sqrt{45} = \sqrt{3^2\cdot5} = 3\sqrt{5}\\ \sqrt{50} = \sqrt{5^2\cdot2} = 5\sqrt{2} $$

Step 2: Multiply the simplified square roots

Now that we have simplified each square root, we can multiply them together: $$ \sqrt{45}\cdot\sqrt{50} = 3\sqrt{5}\cdot5\sqrt{2} $$ This can be further simplified by multiplying the integers together, and then multiply the radicals together: $$ 3\cdot5\cdot\sqrt{5}\cdot\sqrt{2} = 15\sqrt{5}\cdot\sqrt{2} $$

Step 3: Combine the product

Finally, we can combine the square roots into a single square root of their product: $$ 15\sqrt{5\cdot2} = 15\sqrt{10} $$ So, the final answer is: $$ \sqrt{45}\cdot\sqrt{50}=15\sqrt{10} $$

Key Concepts

Prime Factorization

Prime factorization is the process of expressing a composite number as the product of its prime numbers. It is like breaking down a number into the basic building blocks that multiply together to give the original number. For example, when we prime factorize 45, we decompose it into prime factors as follows: 45 can be expressed as \( 9 \times 5 \). Since 9 is not a prime number, we break down 9 further into \( 3 \times 3 \), which each 3 is prime. Thus, 45 can be written as \( 3^2 \times 5 \). Similarly, for 50, we can break it down into \( 25 \times 2 \), where 25 is further factored into \( 5 \times 5 \). Hence, the prime factorization of 50 is \( 5^2 \times 2 \). This method is helpful for simplifying expressions like square roots, making it a fundamental skill in algebra.

Simplifying Radicals

Simplifying radicals involves reducing a radical expression to its simplest form. It typically means making the number inside the square root as small as possible but equivalent in value. For the expression \( \sqrt{45} \), using its prime factorization \( 3^2 \times 5 \), we take the square root of \( 3^2 \) out of the radical, resulting in \( 3\sqrt{5} \). Similarly, \( \sqrt{50} \), using its prime factors \( 5^2 \times 2 \), simplifies to \( 5\sqrt{2} \).

• When simplifying, always look for perfect squares, these are numbers that give whole numbers when square rooted, such as \( 4, 9, 16, 25 \), etc.
• Taking a number squared, like \( 3^2 \) or \( 5^2 \), out of a square root gives a simpler expression.

Simplifying radicals is key in making complex expressions easier to work with, especially when further operations like multiplication are needed.

Multiplying Radicals

When multiplying radicals, the process involves multiplying the numbers outside the radicals together and then multiplying the numbers inside the radicals together. Let's look at the expression \( \sqrt{45} \cdot \sqrt{50} \) which simplifies to \( 3\sqrt{5} \cdot 5\sqrt{2} \) after simplification. To multiply these, first multiply the outside numbers, which are \( 3 \) and \( 5 \), giving \( 15 \). Then, multiply the radicals together: \( \sqrt{5} \cdot \sqrt{2} = \sqrt{5 \cdot 2} = \sqrt{10} \). So, the product becomes \( 15\sqrt{10} \).

• Always simplify each radical before multiplying to make calculations easier.
• The product of the radicals can be combined into a single radical if possible.
• Check if the final expression can be simplified further, such as if \( \sqrt{10} \) could potentially have a perfect square factor.

Multiplying radicals may seem daunting at first, but by breaking the process down step by step, it can become straightforward.