Problem 31 - Prealgebra

Question

Indicate whether each of the statements is True or False. $$\sqrt{25 \cdot 9}=\sqrt{25} \cdot \sqrt{9}$$

Step-by-Step Solution

Answer

True, the equation is correct.

Step 1: Understanding the Statement

The problem asks us to determine whether the equation \( \sqrt{25 \cdot 9} = \sqrt{25} \cdot \sqrt{9} \) is true or false. This will require us to understand and apply properties of square roots.

Step 2: Applying Square Root Properties to the Left Side

The left side of the equation is \( \sqrt{25 \cdot 9} \). According to the property of square roots, \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \) as long as \( a \) and \( b \) are non-negative. Since 25 and 9 are non-negative, the property holds and \( \sqrt{25 \cdot 9} = \sqrt{25} \cdot \sqrt{9} \).

Step 3: Evaluating \( \sqrt{25} \) and \( \sqrt{9} \)

Let's evaluate \( \sqrt{25} \) and \( \sqrt{9} \) separately. We find \( \sqrt{25} = 5 \) and \( \sqrt{9} = 3 \).

Step 4: Multiplying the Evaluated Results

Now multiply the results from the previous step: \( 5 \cdot 3 = 15 \).

Step 5: Conclusion

Both sides of the original equation equal 15. Therefore, the equation \( \sqrt{25 \cdot 9} = \sqrt{25} \cdot \sqrt{9} \) is true.

Key Concepts

Properties of Square Roots

Square roots are mathematical functions often encountered in different algebraic expressions. To understand their properties fully, one should recognize a few foundational rules that apply to non-negative numbers.

• The square root of a product, such as \( \sqrt{a \cdot b} \), can be separated into the product of two square roots: \( \sqrt{a} \cdot \sqrt{b} \), as long as both \( a \) and \( b \) are non-negative. This is known as the product property of square roots.
• This property ensures that for non-negative values, the square root operation can be distributed over a multiplication operation.
• Additionally, one must ensure that both components of the product under the square root remain non-negative to avoid complex numbers.

These properties are extremely helpful, especially in simplifying complex expressions and solving equations efficiently.

Multiplication of Square Roots

Multiplying square roots is a straightforward process once you grasp properties of square roots. If you have \( \sqrt{a} \) and \( \sqrt{b} \), you can multiply these by first multiplying the numbers under the roots: \( \sqrt{a \cdot b} \).

• As shown in our original example, \( \sqrt{25} \cdot \sqrt{9} = \sqrt{25 \cdot 9} \).
• This works due to the property \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).
• It's crucial that the values remain non-negative, keeping calculations straightforward and real.

This property not only simplifies many algebraic expressions but also helps verify the equality in calculations involving square roots.

Evaluating Square Roots

Evaluating square roots means finding the value that, when multiplied by itself, gives the number under the square root. It makes solving equations feasible by simplifying them into more manageable numbers. For instance:

• In our example, evaluating the square roots of 25 and 9 gives us \( \sqrt{25} = 5 \) and \( \sqrt{9} = 3 \).
• Once evaluated, these numbers can be further used in algebraic expressions or calculations. For example, \( 5 \cdot 3 = 15 \).
• This step not only simplifies expressions but also reaches exact numerical answers necessary in solving mathematical problems.

Understanding how to evaluate square roots is a fundamental skill needed to tackle more complex mathematical operations effectively.