Problem 82 - Algebra 1: Concepts and Skills

Question

Write the radical expression in simplest form. $$ 3 \sqrt{63} \cdot \sqrt{4} $$

Step-by-Step Solution

Answer

The simplified form of the given radical expression is \(126\).

Step 1: Simplify the square roots

Start by simplifying individual square roots. The square root of \(63\) is \(7\sqrt{9}\) and the square root of \(4\) is \(2\).

Step 2: Further Simplify the Square Roots

The square root of \(9\) is \(3\), yielding \(7 \cdot 3\) which equals \(21\). Hence, \(3\sqrt{63}\) simplifies to \(3 \cdot 21\).

Step 3: Multiply the Constants

Finally, perform the multiplication operation as indicated in the radical expression, obtaining \(3 \cdot 21 \cdot 2\).

Step 4: Compute the Operation

Perform the multiplication to get the final answer. The result is \(126\).

Key Concepts

Square Roots

The concept of square roots is fundamental in mathematics, especially in simplifying radical expressions. A square root is a number that, when multiplied by itself, gives the original number. For instance, the square root of 4 is 2 because 2 multiplied by itself equals 4. Square roots are often seen in the form of the radical symbol \( \sqrt{} \). When simplifying square roots like \( \sqrt{63} \), you look for factors that are perfect squares. Here, 63 can be broken down into \( 9 \times 7 \), where 9 is a perfect square. Thus, \( \sqrt{63} = \sqrt{9 \times 7} = 3\sqrt{7} \). Breaking numbers down to their smallest components helps in simplifying expressions significantly.
When square roots are multiplied, such as \( \sqrt{63} \cdot \sqrt{4} \), the property \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \) can be applied, making simplification possible. Understanding these properties helps in tackling more complex algebraic expressions with ease.

Multiplication in Algebra

Multiplication is a key operation in algebra that involves combining terms or expressions. When working with radicals, it's crucial to manage constants and variables carefully. Suppose you have an expression like \( 3 \sqrt{63} \cdot \sqrt{4} \). Here, you can first address the radicals: \( \sqrt{63} \cdot \sqrt{4} = \sqrt{63 \cdot 4} \). This operation makes use of the multiplication property of square roots, which states that the product of square roots is equal to the square root of the product.
Next, you'll perform the arithmetic operation by first simplifying any constants or coefficients in front of the radicals. For instance, multiplying 3 by the simplified square root of 63 gives \( 3 \cdot 21 \), since \( \sqrt{63} \) can be simplified to \( 3 \sqrt{7} \). Multiplying constants follows the regular rules, where

• The commutative property applies, meaning the order does not affect the result: \( a \cdot b = b \cdot a \).
• Multiplying these constants gives you the expression \( 3 \cdot 21 \cdot 2 \) to further solve.

Mastering multiplication in algebra allows you to more effectively break down and simplify complex expressions.

Simplifying Expressions

Simplifying expressions is an important skill in algebra that involves reducing expressions to their simplest forms. The process typically involves combining like terms, reducing coefficients, and handling any radicals appropriately.
To simplify the expression \( 3 \sqrt{63} \cdot \sqrt{4} \), begin with breaking down each radical into its simplest form. As seen in the solution, \( \sqrt{63} \) simplifies to \( 3 \sqrt{7} \) and \( \sqrt{4} \) simplifies to 2. Then, you multiply the coefficients: \( 3 \times 21 \times 2 \) results in \( 126 \).
Here's a step-by-step approach to simplifying expressions:

• Simplify inside any radicals first to reduce complexity. Look for perfect squares as factors.
• Multiply constants or coefficients separately from variables and radicals.
• Combine all results to obtain the simplest form of the expression.

By following these steps, you ensure that your expressions are not just simpler, but also easier to handle in further calculations.