Problem 84 - Algebra 1: Concepts and Skills

Question

Write the radical expression in simplest form. $$ \sqrt{9} \cdot 4 \sqrt{25} $$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt{9} \cdot 4 \sqrt{25}\) is 60.

Step 1: Simplify Radicands

First, simplify the radicands. Square root of 9 is 3 and square root of 25 is 5.

Step 2: Apply the Product Property

Second, apply the product rule. The product rule states that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\). However, in this case, both radicands are already simplified, thus the rule can be applied inversely. So, instead of multiplying under the radical, you multiply the results from the first step, which are the square roots of each radicand.

Step 3: Calculate the Result

Finally, calculate the result. The product 3 (from \(\sqrt{9}\)) times 4 times 5 (from \(4 \sqrt{25}\)) equals 60.

Key Concepts

Understanding Square Roots

Square roots are a foundational concept in mathematics, representing a number that, when multiplied by itself, gives the original value under the square root. For instance, \( \sqrt{9} \) is 3, because 3 times 3 equals 9. Similarly, \( \sqrt{25} \) is 5, because 5 times 5 equals 25. It's helpful to remember that:

• The square root of a perfect square like 9 or 25 is always a whole number.
• Square roots can be simplified if the number inside the square root (radicand) is a perfect square.
• If the radicand isn’t a perfect square, it might be simplified by breaking it down into factors.

This simplification makes handling and calculating with radicals much more efficient and straightforward.

Exploring the Product Property of Radicals

The product property of radicals is an essential concept for simplifying expressions involving square roots. It states that:\[\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\]
This property allows us to multiply under the same radical sign or apply it inversely for simplification. In the context of our problem, even though the radical expressions \( \sqrt{9} \) and \( 4\sqrt{25} \) are not multiplied directly under one radical, knowing how the property works helps us simplify without confusion:

• You can multiply the simplified results of radicals as in: 3 (from \( \sqrt{9} \)) and 5 (from \( \sqrt{25} \)).
• In this case, multiplying those results by 4 gives us the correct simplified expression.

The product property of radicals is particularly useful when dealing with multiple radical expressions in a single equation.

Simplifying Radicands for Easier Calculations

Simplifying radicands is the process of making the number inside the square root easier to handle by breaking it down into simpler components. Here’s how it applies:

• Check if the radicand is a perfect square, like 9 or 25, which can immediately be simplified to whole numbers (3 and 5).
• If it's not a perfect square, decompose it into factors, looking for perfect squares within those factors.
• Doing so can make subsequent calculations more manageable and less error-prone.

By simplifying radicands, you significantly reduce the complexity of a problem, making it easier to apply further steps or properties, such as the product property of radicals, to get to the final answer seamlessly.