Problem 44 - Algebra 1

Question

Simplify the expression. $$\sqrt{9} \cdot 4 \sqrt{25}$$

Step-by-Step Solution

Answer

The simplified expression is 60

Step 1: Evaluate Square Roots

Start by evaluating the square roots. The square root of 9 is 3, and the square root of 25 is 5.

Step 2: Substitute into the Original Expression

Next, substitute the evaluated square roots back into the original expression. This gives \(3 * 4 * 5\).

Step 3: Simplify the Expression

Multiply the numbers together to simplify the expression. \(3 * 4 * 5 = 60\).

Key Concepts

Square Root Evaluation

Square roots are mathematical operations that help us find a number which when multiplied by itself yields the original number. For instance, finding the square root of 9 means finding some number that, when multiplied by itself, gives 9. In this case, that number is 3 since, \(3 \times 3 = 9\). Similarly, the square root of 25 is 5 because \(5 \times 5 = 25\). These calculations help simplify expressions that involve square roots. It's like breaking down a problem into pieces that are easier to manage.

To evaluate a square root, consider the following steps:

• Identify the number under the square root symbol \(\sqrt{}\).
• Determine the number that when multiplied by itself results in the original number.
• If the number is not a perfect square, use a calculator for an approximate value, but for simplification in expressions, exact values are preferred.

Understanding how to evaluate square roots is fundamental, as it sets the foundation for simplifying more complex mathematical expressions.

Multiplication of Radicals

When you deal with multiplication of radicals (expressions with square roots), it's crucial to understand that each radical can be evaluated separately first and then multiplied together. This method makes it easier to simplify the expression correctly. In our example, we have \(\sqrt{9}\) which is 3, and \(4\sqrt{25}\) which is \(4 \times 5 = 20\). These results arise because once you simplify the square roots, the expression becomes a straightforward multiplication of numbers.

The steps to multiply radicals are:

• Evaluate each square root individually first.
• Substitute these evaluated values back into the expression, as we did with \(3\) and \(20\).
• Multiply these numbers together just like you would with any regular numbers.

Remember, simplifying square roots before multiplying makes your process smoother and minimizes errors in calculation.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators (like addition, subtraction, multiplication, etc.). Simplifying these expressions means reducing them to their simplest form, often to make calculations or understanding easier. In our initial example, the expression \(\sqrt{9} \cdot 4 \sqrt{25}\) contained operations that seemed complex at first. By evaluating each square root and simplifying step by step, we turned the expression into \(3 \times 4 \times 5\).

Here’s how you simplify mathematical expressions, particularly those involving square roots:

• Evaluate any square roots first to convert them into regular numbers.
• Perform operations in the correct order, often referred to as "order of operations" or PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).
• Simplify the results by combining all like terms, if any, into a single final number.

Simplifying expressions not only helps in finding solutions but also aids in understanding the relationships between different mathematical components. This process is an essential skill for students learning algebra and more advanced math concepts.