Problem 76 - Intermediate Algebra

Question

Evaluate each expression. See Example \(9 .\) $$ |\sqrt{49}-8(4-7)| $$

Step-by-Step Solution

Answer

The value of the expression is 31.

Step 1: Evaluate the Square Root

The expression starts with a square root. We need to calculate \( \sqrt{49} \) first. Recall that 49 is a perfect square, and its square root is 7. So \( \sqrt{49} = 7 \).

Step 2: Evaluate the Parentheses

Within the expression, the next operation is inside the parentheses: \( 4 - 7 \). Subtract 7 from 4 to get \( 4 - 7 = -3 \).

Step 3: Evaluate the Multiplication

Next, multiply \(-3\) by 8 as indicated by the expression. \(-3 \times 8 = -24\).

Step 4: Perform the Subtraction

Now, subtract the result from Step 3 (which is -24) from the square root result, which is \(7 - (-24)\). Specifically, this becomes \(7 + 24 = 31\) because subtracting a negative number is equivalent to adding its positive counterpart.

Step 5: Absolute Value

Finally, remember the expression is within an absolute value: \(|31|\). The absolute value of 31 is just 31, since it is already a positive number.

Key Concepts

Understanding Square Roots

Square roots are fundamental in math. They can seem tricky, but with some explanation, they become clearer. When we take the square root of a number, we're looking for a value that, when multiplied by itself, gives us the original number.
For instance, the square root of 49 is 7, because 7 multiplied by 7 is 49. It’s like finding out what times itself equals the number we started with. This is why square roots are often associated with perfect squares.

• Perfect Squares: Numbers like 49, where the square root is an even whole number, are known as perfect squares.
• Symbol: We denote square roots with the symbol \(\sqrt{}\).

Identifying perfect squares can simplify problems, especially in expressions like in our example.

Decoding Order of Operations

The order of operations is like the rulebook for math calculations. It ensures everyone evaluates mathematical expressions the same way, avoiding errors.
PEMDAS, or BIDMAS in some parts of the world, guides us:

• Parentheses: Deal with any calculations inside parentheses first.
• Exponents: Next, handle any exponents or powers.
• Multiplication/Division: Then, proceed with any multiplication and division, from left to right.
• Addition/Subtraction: Finally, perform any addition and subtraction, also from left to right.

In the given example, we follow this order to solve \(|\sqrt{49} - 8(4-7)|\). Parentheses are tackled first, simplifying \(4-7\), then the rest follows, ensuring accuracy.

Exploring Perfect Squares

Perfect squares make square root operations smoother. A perfect square is simply the result of squaring a whole number.
Taking 7 as an example, squaring it gives us 49, making 49 a perfect square. Perfect squares are like reliable friends in math—they make calculations neat by being predictable.

• Common Examples: 1, 4, 9, 16, 25, 36, 49, etc., are all perfect squares.
• Usage: Recognizing them quickly can save time, especially in manual calculations.

Using perfect squares helps in swiftly identifying the square root in expressions, eliminating guesswork.

Evaluating Expressions with Absolute Values

Evaluating expressions involves systematically simplifying them using various mathematical rules. One such rule is the operation involving absolute values.
Absolute value, represented by vertical bars like \(|x|\), gives the non-negative equivalent of a number. It's like asking, "How far is this number from zero?"

• Application: Absolute values turn negative numbers positive while keeping positive numbers the same.
• Real-world Analogy: Think of it as measuring distance—distance is always positive regardless of direction.

In our expression \(|31|\), the number is already positive. Thus, its absolute value remains 31, demonstrating how absolute value simplifies results by neutralizing negatives.