Problem 29 - Prealgebra

Question

Indicate whether each of the statements is True or False. $$\sqrt{4}+\sqrt{9}=\sqrt{4+9}$$

Step-by-Step Solution

Answer

False

Step 1: Evaluate the Left Side of the Equation

First, let's evaluate the left side of the equation: \( \sqrt{4} + \sqrt{9} \). We need to find the square roots of 4 and 9. The square root of 4 is 2, and the square root of 9 is 3. Therefore, the left side of the equation becomes: \( 2 + 3 = 5 \).

Step 2: Evaluate the Right Side of the Equation

Now, let's evaluate the right side of the equation: \( \sqrt{4+9} \). We add the numbers inside the square root first, which gives us \( 4 + 9 = 13 \). Thus, we have \( \sqrt{13} \) on the right side of the equation.

Step 3: Compare the Two Sides of the Equation

Compare the values from the previous steps. The left side of the equation is 5, and the right side is \( \sqrt{13} \). Since 5 is not equal to \( \sqrt{13} \), the equation \( \sqrt{4} + \sqrt{9} = \sqrt{4+9} \) does not hold.

Key Concepts

Evaluating Expressions

When you're evaluating expressions, you're essentially finding the value of an expression by applying mathematical operations. Let's break it down with the given equation, \(\sqrt{4} + \sqrt{9} = \sqrt{4+9}\). Your task is to find out if this statement is true or false by evaluating each side.

Here's how you can do it:

• First, take the square root of each term separately on the left side: \(\sqrt{4}\) becomes 2, and \(\sqrt{9}\) becomes 3.
• Next, add these two results: 2 + 3 = 5.
• On the right side, sum the numbers inside the square root first: 4 + 9 equals 13, and thus you have \(\sqrt{13}\).

Now compare the results. The left side equals 5, while the right side gives you \(\sqrt{13}\).

If you know that 5 is not the same as the approximate value of \(\sqrt{13}\) (which is around 3.6), you've correctly evaluated the expression to see the differences on both sides. This is an essential skill for solving math problems accurately.

Square Root Properties

Square root properties play an important role in simplifying and evaluating mathematical expressions. A crucial property to remember is that the square root of a sum is not equal to the sum of the square roots.

This means \(\sqrt{a+b}\) is not the same as \(\sqrt{a} + \sqrt{b}\). You saw this property in action with our exercise.

Here's why:

• The square root function is not linear, which means you can't just separate terms under a radical into separate square roots.
• The value of \(\sqrt{13}\) is not 5, which would be the sum of the roots, \(\sqrt{4}\) and \(\sqrt{9}\).

Understanding these properties lets you avoid common errors when dealing with square roots in algebra and arithmetic. Always evaluate operations inside the square root fully before taking the root itself.

True or False Math Statements

Deciding whether math statements are true or false requires careful evaluation and understanding of mathematical properties. In our example, the statement \(\sqrt{4} + \sqrt{9} = \sqrt{4+9}\) is false.

Here's the verification process:

• We evaluated each side separately, with the left being 5 and the right approximately 3.6 (since \(\sqrt{13}\) is close to 3.6).
• By comparing results, you see that 5 and 3.6 are different, showing the statement cannot be true.

Remember, always

• Ensure calculations follow the correct order of operations and properties.
• Use logical reasoning to identify mistakes in equations or assumptions.

This skill is crucial and transfers across many areas of math, from simple arithmetic to complex algebra.