Problem 2 - Elementary Algebra

Question

Find each of the following products. $$ \sqrt{32} \sqrt{2} $$

Step-by-Step Solution

Answer

Answer: The product is 8.

Step 1: Multiply the square roots

Using the property of multiplication of square roots, we can multiply \(\sqrt{32}\) and \(\sqrt{2}\) like this: $$ \sqrt{32} \cdot \sqrt{2} = \sqrt{32 \times 2} $$

Step 2: Calculate the product inside the square root

Now, we need to find the product of 32 and 2: $$ 32 \times 2 = 64 $$ Replace the product in the square root: $$ \sqrt{32 \times 2} = \sqrt{64} $$

Step 3: Simplify the square root

Lastly, we need to find the square root of 64: $$ \sqrt{64} = 8 $$ So, the product of \(\sqrt{32}\) and \(\sqrt{2}\) is 8.

Key Concepts

Multiplication of Square Roots

When multiplying square roots, you can use the property that says: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). This property allows us to simplify the process of multiplication by combining the expressions under a single square root. For example, in the problem \( \sqrt{32} \cdot \sqrt{2} \), we apply this property to combine them into \( \sqrt{32 \times 2} \).
By understanding this property, you simplify your calculations significantly, especially when dealing with larger numbers under the square root.
Here is how it works:

• Multiply the numbers inside the square roots together.
• Place the product under one square root.

This method is very useful and keeps the calculations straightforward, which makes handling square roots more approachable.

Simplifying Square Roots

Simplifying square roots is about breaking down the square root into its simplest form. Once you've multiplied the square roots and combined them into one expression, as \( \sqrt{64} \) in our example, the next step is to simplify. To do so, we determine the square root of that number.
In our exercise, \( \sqrt{64} = 8 \). Knowing the perfect squares, such as 64, helps in instantaneous simplification.
If you encounter a non-perfect square:

• Find the prime factorization of the number.
• Pair the prime factors.
• Bring one factor from each pair outside the square root.

This method ensures you simplify every square root to the lowest possible form, making it easier to work with further.

Properties of Square Roots

There are some key properties of square roots that are fundamental to solving mathematical problems. Understanding these properties makes working with square roots easier and more intuitive. Three vital properties include:

• Multiplicative Property: As mentioned earlier, \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). This is particularly useful in simplifying expressions.
• Additive Property: \( \sqrt{a^2} = a \), where \( a \) is any non-negative number. This property helps in directly computing square roots of perfect squares.
• Non-negative Property: The square root function outputs only non-negative numbers, i.e., \( \sqrt{a} \geq 0 \) for \( a \geq 0 \).

By leveraging these properties, you can effectively navigate through complex mathematical equations involving square roots, ensuring you can simplify calculations and arrive at correct solutions effortlessly.