Problem 46 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ \left[-3(-1)^{3}-4(-2)^{2}\right]^{2} $$

Step-by-Step Solution

Answer

The simplified expression is 169.

Step 1: Evaluate Exponents

Begin by evaluating the exponents in the expression. For \((-1)^3\), this equals \(-1\) because a negative base raised to an odd exponent remains negative. For \((-2)^2\), this equals \(4\) because a negative base raised to an even exponent becomes positive.

Step 2: Simplify Inside Brackets

Substitute the results found in Step 1 back into the expression. You have: \(-3(-1) - 4(4)\). Simplify each term separately: \(-3(-1)\) becomes \(3\) because multiplying two negative numbers yields a positive number. \(-4(4)\) becomes \(-16\) because multiplying negative and positive numbers yields a negative number.

Step 3: Combine Like Terms

Add the results obtained in Step 2. The expression \(3 - 16\) simplifies to \(-13\).

Step 4: Evaluate the Outer Exponent

Now, apply the outer exponent to the simplified term. \((-13)^2\) equals \(169\), because squaring a negative number results in a positive number.

Key Concepts

Exponents

When we talk about exponents, we are referring to a number being multiplied by itself a certain number of times. In mathematics, an exponent is usually denoted by a small number written above and to the right of a base number. For example, in the expression \((-2)^2\), the number \(-2\) is the base, and 2 is the exponent. This means we are multiplying \(-2\) by itself:

• \[(-2) \times (-2) = 4\]

When the base is a negative number and the exponent is:

• Even: The result is positive because a negative number multiplied by itself an even number of times becomes positive.
• Odd: The result is negative, as seen in \((-1)^3\),
  • \((-1) \times (-1) \times (-1) = -1\)
  which simply follows the rule that an odd power of a negative number stays negative.

Remembering these basic rules will always help while simplifying expressions involving exponents.

Negative Numbers

Negative numbers can sometimes confuse students, especially when combined with other operations like multiplication and division. These numbers are not above the zero line of a number line; instead, they are below, representing values less than zero. Here's how they work:

• A negative times a negative results in a positive. Like in our example, \(-3 \, (-1)\), which becomes 3. Imagining this in real-life terms might help: if subtracting a debt makes you richer, then two negatives indeed make a positive.
• A negative times a positive results in a negative. An example is \(-4 \, \times 4\), which gives us \(-16\). It is like advancing in the negative direction on a number line, which takes you further away from zero.

Visualizing on a number line or thinking of everyday scenarios might aid your understanding of why multiplying or dividing negative numbers behaves in this way.

Order of Operations

The order of operations is a key arithmetic rule that dictates the sequence in which the operations should be performed to correctly simplify expressions. To remember the right order, you can use the acronym PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In the original problem, we adhere to this order:

• We first handle the exponents within the brackets.
• Then conduct any multiplication or division.
• We handle operations within the brackets before applying the outer exponent, as operations within brackets take precedence.

Using this systematic approach ensures that mathematical expressions are consistently simplified correctly, leading to accurate results.