Problem 16 - Precalculus

Question

Evaluate each expression. $$\left|(-7)^{2}\right|+|-7|^{2}-(-|-3|)^{3}$$

Step-by-Step Solution

Answer

The expression evaluates to 125.

Step 1: Evaluate the First Absolute Value

Begin by evaluating \( (-7)^2 \). Calculating inside the absolute value, \( (-7)^2 = 49 \). Since 49 is already positive, \( |49| = 49 \).

Step 2: Evaluate the Second Absolute Value

Next, evaluate \(|-7|\). Since -7 is negative, its absolute value is positive: \(|-7| = 7\). We then find \(7^2 = 49\).

Step 3: Evaluate the Last Expression

Consider the expression \(-|-3|\). First, find \(|-3|\) which is \(3\). Then, its negative is \-3\. Now calculate \((-3)^3\): \(-3 imes -3 imes -3 = -27\).

Step 4: Calculate the Final Expression

Now combine all parts together: \(49 + 49 - (-27)\). Simplify by evaluating the terms step-by-step: \(49 + 49 = 98\), and \(98 - (-27) = 98 + 27 = 125\).

Key Concepts

Understanding Exponents

Exponents indicate how many times a number, known as the base, is multiplied by itself. The expression \((-7)^2\) implies that \(-7\) is multiplied by itself once: \((-7) \times (-7)\). Here's how you handle expressions with exponents:

• Determine the base and exponent. For \((-7)^2\), \(-7\) is the base and 2 is the exponent.


• Multiply the base by itself according to the number of the exponent. So, \((-7)^2 = (-7) \times (-7) = 49\).


• Pay attention to negative signs. A negative base raised to an even exponent results in a positive number (like 49). A negative base raised to an odd exponent keeps the negative sign.

Understanding exponents is essential for evaluating expressions accurately, as you can see in the provided exercise where we calculated \((-7)^2\) as 49.

Mastering the Order of Operations

The order of operations is crucial for solving mathematical equations correctly. It ensures consistency, allowing different people to evaluate expressions and arrive at the same answer. The sequence is often remembered by the acronym PEMDAS:

• Parentheses


• Exponents


• Multiplication and Division (from left to right)


• Addition and Subtraction (from left to right)

When evaluating expressions like the one in the exercise, follow PEMDAS:First, handle operations inside any parentheses or absolute values, as they work like parentheses. In our expression, we first calculated inside the absolute values of \((-7)^2\) and \(-|-3|\). Next, we moved on to clear exponents such as \(7^2\). Only then did we proceed to add and subtract the resolved terms. This step-by-step adherence to the order of operations ensures each part of the problem is addressed accurately.

Integer Operations Simplified

Integer operations involve arithmetic on whole numbers which include positive numbers, negative numbers, and zero. They are vital for understanding expressions like the given example:

• Absolute Values: This represents the non-negative value of a number. That is, \|-7| = 7\and \|-3| = 3\. These operations return the distance of a number from zero.


• Negative and Positive Integer Operations: When subtracting, remember that subtracting a negative is the same as addition. That's why \(98 - (-27)\) becomes \(98 + 27\).


• Multiplication and Exponents: Be mindful of the sign when multiplying or raising integers to a power. For the expression \((-3)^3\), it resulted in \-27\ because multiplying an odd number of negative numbers results in a negative product.

By mastering integer operations, you can confidently evaluate the arithmetic parts of more complex mathematical expressions. This knowledge is applied in our exercise, especially when calculating absolute values and handling signs during addition and subtraction.