Problem 38 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ 5(-1)^{3}-(-3)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is 22.

Step 1: Simplify the Exponentiated Terms

First, calculate the value of \((-1)^3\) and \((-3)^3\).\[(-1)^3 = -1 \times -1 \times -1 = -1\]\[(-3)^3 = -3 \times -3 \times -3 = -27\]

Step 2: Apply the Negative Sign

Now, account for the negative sign in front of the \((-3)^3\) term.Since there is a negative sign in front of \((-3)^3\), we have:\(-(-3)^3 = -(-27) = 27\).

Step 3: Substitute Back into the Expression

Substitute the simplified values back into the original expression:\[5(-1)^3 - (-3)^3 = 5(-1) + 27\]

Step 4: Calculate the Result

Now, multiply the 5 and \(-1\): \[5(-1) = -5\] Then perform the addition:\(-5 + 27 = 22\)

Key Concepts

Exponentiation

Exponentiation is the mathematical operation where a number, known as the base, is multiplied by itself a specified number of times, called the exponent. For example, in the expression \((-1)^3\),

• the base is \(-1\)
• the exponent is 3

This means that \((-1)\) should be multiplied by itself 3 times, resulting in: \[ (-1) \times (-1) \times (-1) = -1 \]When dealing with odd exponents and negative bases, the result retains the negative sign. In contrast, negative bases raised to an even exponent will result in a positive product.

• For example, \((-2)^2 = 4\)
• Whereas \((-2)^3 = -8\)

Understanding this behavior helps in correctly simplifying expressions involving exponents.

Simplifying Expressions

Simplifying expressions often involves breaking them down into more manageable parts, making calculations more straightforward. In our example, we have the expression \(5(-1)^3 - (-3)^3\).

• We start by addressing exponentiation, as this is fundamental before other operations.
• Once exponentiated components are resolved, move to arithmetic operations such as addition or subtraction.

In our process:

• \((-1)^3\) was simplified to \(-1\)
• \((-3)^3\) became \(-27\)

Next, handling any negative signs correctly is crucial. Negative signs in front of exponential terms can change how they are combined. After simplifying the exponents, substitute them back into the original expression. Finish by performing the straightforward arithmetic to achieve the final simplification like this:

• Substitution gives \(5(-1) + 27\)
• Finally simplifies to \(-5 + 27 = 22\)

These steps showcase the importance of a systematic approach in simplifying numerical expressions.

Negative Numbers

Negative numbers, those less than zero, exhibit unique properties that can affect outcomes in arithmetic operations. When working with expressions that include negative numbers, special attention is needed:

• Multiplying two negative numbers results in a positive number, e.g., \((-2) \times (-3) = 6\)
• Multiplying a negative number by a positive number keeps the result negative, e.g., \((-2) \times 3 = -6\)

In the exercise, negative numbers were primarily present in the bases of exponents. Notice how influencing factors like the exponent also determine the overall sign:

• \((-1)^3\) maintained its negative value, while
• \(-(-3)^3\) inverted the sign to positive due to the initial subtraction sign.

Managing sign changes is a crucial skill when simplifying expressions involving negative numbers. It's often helpful to visualize the process or write down intermediary sign results to avoid mistakes.