Problem 54 - Prealgebra

Question

Compute the exact value of the given expression. \(-2-7 \sqrt{576}\)

Step-by-Step Solution

Answer

The exact value of the expression is -170.

Step 1: Simplify the square root

The first part of the expression is the square root. We know that the square root of a number is a value that, when multiplied by itself, gives the original number. We need to simplify the square root of 576. We see that \( 576 = 24 \times 24 \), hence \( \sqrt{576} = 24 \).

Step 2: Substitute the simplified value

Now substitute the value of \( \sqrt{576} \) back into the expression. Therefore, the expression becomes: \(-2 - 7 \times 24\).

Step 3: Perform multiplication

Now, multiply 7 by 24. The result is: \( 7 \times 24 = 168 \).

Step 4: Apply subtraction

Substitute the product from Step 3 back into the equation and perform the subtraction: \(-2 - 168 = -170 \).

Key Concepts

Square Roots

Understanding square roots is crucial for simplifying expressions, especially when dealing with real numbers. A square root asks: what number multiplied by itself equals the target number? For example, the square root of 576 is found by determining which number squared equals 576. Rewriting 576 as a product of two identical factors, we have:

• 576 = 24 \times 24
• \(\sqrt{576} = 24\)

This shows that 24 is the number that, when multiplied by itself, results in 576. Recognizing this step helps simplify complex expressions by replacing the square root with a whole number. This simplification plays a key role in making the original mathematical operations easier to handle.

Multiplication

In mathematics, multiplication is a way of combining equal groups. It's crucial for evaluating expressions efficiently. Once you have simplified the square root, the next step is solving the multiplication part of the expression, such as \(-2 - 7 \times 24\).

• First, multiply 7 by 24.
• You get 7 \times 24 = 168.

Breaking down the multiplication step ensures that each part of the expression is clearly understood. Multiplying accurately is essential for maintaining the correct sequence and value in the overall expression. This leads us into the final operation of subtraction.

Subtraction

Subtraction is one of the basic arithmetic operations, used to find the difference between numbers. In mathematical expressions, it’s crucial to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). After handling square roots and multiplication, subtraction finalizes the computation.

• Begin with the expression \(-2 - 168\).
• Subtract 168 from -2.
• This results in \( -170 \).

Thus, subtraction not only reduces the expression to its final form but also ensures the accuracy of the solution. Each step confirms that arithmetic rules and operations are applied correctly, leading to the precise value of the expression.