Problem 8 - Elementary and Intermediate Algebra

Question

Multiply. $$ \sqrt{10} \sqrt{3} $$

Step-by-Step Solution

Answer

The result is \( \sqrt{30} \).

Step 1: Identify the Multiplicands

Recognize that we need to multiply two square roots: \ \ \( \sqrt{10} \) and \( \sqrt{3} \).

Step 2: Use the Property of Square Roots

Apply the property of square roots which states that \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). Here, \( a = 10 \) and \( b = 3 \).\ \ Therefore, \( \sqrt{10} \cdot \sqrt{3} = \sqrt{10 \cdot 3} \).

Step 3: Perform the Multiplication Inside the Square Root

Calculate the product inside the square root: \ \ \( 10 \cdot 3 = 30 \).\ This simplifies the expression to: \ \( \sqrt{30} \).

Key Concepts

Properties of Square Roots

Square roots have several properties that make them easier to work with, especially when multiplying or dividing them. One of the most useful properties is the product rule, which states:
\[\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\]
This means that you can multiply the numbers inside the square roots first, and then take the square root of the resulting product. For example, when multiplying \(\sqrt{10}\) and \(\sqrt{3}\), you can combine them inside one square root:
\( \sqrt{10} \cdot \sqrt{3} = \sqrt{10 \cdot 3} = \sqrt{30} \).
This property simplifies calculations and is very helpful in algebra and dealing with radical expressions.

Radical Expressions

Radical expressions involve roots, such as square roots. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, \( \sqrt{9} = 3 \) because \( 3^2 = 9 \).
Radical expressions can include numbers, variables, or both under the square root symbol. When dealing with radical expressions, it's important to know how to simplify and manipulate them using different properties, including the multiplication property we used earlier. This helps to handle more complex algebraic expressions efficiently.

Simplifying Radicals

Simplifying radicals means making the expression under the square root as simple as possible. Often, this involves breaking numbers down into their prime factors. For instance, to simplify \(\sqrt{30}\), you would break down 30 into its prime factors: \( 30 = 2 \cdot 3 \cdot 5 \).
If a number is a perfect square (like 4, 9, 16), its square root is an integer. However, in many cases, such as \( \sqrt{30} \), it is already in its simplest form because 30 does not have any perfect square factors. Simplifying radicals helps in keeping expressions clear and concise, which is particularly useful for solving equations and other algebraic tasks.