Problem 32 - Prealgebra

Question

Indicate whether each of the statements is True or False. $$\sqrt{100}-\sqrt{36}=\sqrt{100-36}$$

Step-by-Step Solution

Answer

The statement is False.

Step 1: Calculate the Left Side of the Equation

First, calculate the square roots on the left side of the equation: \( \sqrt{100} = 10 \) and \( \sqrt{36} = 6 \). Therefore, \( \sqrt{100} - \sqrt{36} = 10 - 6 = 4 \).

Step 2: Calculate the Right Side of the Equation

Now calculate the square root on the right side of the equation. First, subtract inside the square root: \( 100 - 36 = 64 \). Then take the square root: \( \sqrt{64} = 8 \).

Step 3: Compare Both Sides

Now compare the results from both sides: the left side gives us 4, and the right side gives us 8. Since 4 does not equal 8, the statement is false.

Key Concepts

Understanding Square Roots

Square roots are an important part of mathematics, especially when dealing with real numbers. The square root of a number, say \( x \), is a value that when multiplied by itself gives \( x \). For example, \( \sqrt{100} = 10 \) because \( 10 \times 10 = 100 \). Square roots are fundamental to equations and expressions, allowing us to simplify and solve them effectively.

It's crucial to recognize that square roots only deal with non-negative numbers in the real number domain. When you see \( \sqrt{36} \), it refers to 6, not -6, as we often use the principal (non-negative) square root.

• Always remember: \( \sqrt{x^2} = |x| \), ensuring results are non-negative.
• Practicing with square roots improves problem-solving skills, especially when dealing with complex equations.

Solving Equations Using Square Roots

Equations often involve operations with square roots, demanding precision in calculations. Solving these equations requires understanding the relationship between different terms. Let's look at the exercise with both sides of the equation: - On the left, we have: \( \sqrt{100} - \sqrt{36} \) which simplifies to \( 10 - 6 = 4 \). - On the right, it's \( \sqrt{100 - 36} \). Subtracting gives \( 64 \), and its square root is \( 8 \).

When solving these equations, be mindful of order and operations. Missteps can easily lead to incorrect results.

• Perform operations within the square roots first, then resolve any other arithmetic.
• Compare your results carefully to understand if the equation holds true.

Verifying True or False Statements

Mathematical statements, like the one in the exercise, often require verification. The given statement involved comparing two expressions: \( \sqrt{100} - \sqrt{36} \) and \( \sqrt{100 - 36} \). To determine if a statement is true or false, you break it down into smaller, manageable calculations.

Start by solving each side separately, as shown earlier, then compare the results.

• If both sides are equal, the statement is true.
• If they are not, it's false, just like the expression \( 4 eq 8 \) in this exercise.

Taking methodical steps ensures a comprehensive understanding and avoids errors. Developing these skills is great for analytical thinking and building robust mathematical reasoning.