Problem 39 - Fundamentals of Mathematics

Question

Determine each of the values. $$ -\left\\{-[-|-4|+|-3|]^{3}\right\\}^{2} $$

Step-by-Step Solution

Answer

117649

Step 1: Understand the Absolute Values

Start by identifying and simplifying the absolute values in the expression. Within the expression \( |-4| + |-3| \), evaluate each absolute value separately. \( |-4| = 4 \) and \( |-3| = 3 \) because the absolute value of a number is its non-negative counterpart.

Step 2: Simplify Inside the Parentheses

Substitute the simplified absolute values back into the expression. Thus, the expression becomes \( -[-(4+3)]^{3} \). Add the numbers inside the parentheses: \( 4 + 3 = 7 \).

Step 3: Evaluate the Negative Sign Inside the Parentheses

Next, remove the inner negative sign by evaluating \( -7 \). This gives \( -(7) = -7 \).

Step 4: Raise to the Power of 3

Raise \( -7 \) to the power of 3. \((-7)^3 = -7 \times -7 \times -7 = -343\).

Step 5: Address the Outer Negation and Square

Next, remove the outer negative sign by making \(( -343 )\) positive and square it. The expression becomes \( (-1 \cdot -343)^2 = 343^2 \). Then calculate \((343)^2 = 117649\).

Key Concepts

Evaluating Expressions

Evaluating expressions is a process of replacing variables with actual numbers, carefully following the sequence of mathematical operations to find a numerical result. An expression can include numbers, variables, and different operations like addition, subtraction, multiplication, division, and exponents.

For example, consider an expression involving absolute values and other operations, such as:

• An absolute value, which determines how far a number is from zero on a number line, always resulting in a non-negative number.
• Each segment of the expression is addressed in a specific order. In this case, absolute values must be calculated first.
• Once simplified, we can substitute them back into the expression for further calculations.

By following these steps methodically, one can accurately evaluate any expression, even when it includes nested operations like parentheses and exponents.

Exponents

Exponents represent repeated multiplication of a number by itself, and are a key concept in algebra. Understanding how to work with exponents is crucial for simplifying expressions. The exponent tells you how many times to multiply the base by itself.

For example, in our problem, raising \[-7^3\] means multiplying \(-7\) by itself twice more: \[-7 \times -7 \times -7 = -343\].

• The base here is \(-7\) and the exponent is \(3\).
• Negative bases raised to an odd exponent keep their negativity, whereas raising them to an even exponent results in a positive number.
• Thus, careful attention to positive and negative signs in combination with exponents is important in solving algebraic expressions accurately.

Understanding exponents not only helps in solving algebra problems, but also in grasping more complex mathematical concepts related to growth, scaling, and scientific notation.

Order of Operations

The order of operations is a fundamental principle in mathematics that dictates the sequence in which operations should be carried out to ensure accurate results. Following this order is vital to accurately evaluate expressions.

The order is commonly remembered using the acronym PEMDAS:

• **P**: Parentheses first
• **E**: Exponents (powers and roots)
• **MD**: Multiplication and Division (left to right)
• **AS**: Addition and Subtraction (left to right)

Applying PEMDAS ensures that all expressions are simplified correctly. For instance, when solving \(-\left\{-[-|-4|+|-3|]^3\right\}^2\), you would first deal with absolute values and parentheses, followed by exponents, before addressing multiplication or division, addition, or subtraction.

Incorrectly applying the order of operations can lead to errors, thus understanding and practicing PEMDAS is crucial for mathematical success. It forms the foundation for more advanced problem-solving skills.