Problem 53 - Algebra A Combined Function

Question

Perform the indicated operation. \((-5)^{3}\)

Step-by-Step Solution

Answer

The result of \((-5)^3\) is \(-125\).

Step 1: Identify the Problem

We need to evaluate the expression \((-5)^3\). This means multiplying \(-5\) by itself three times.

Step 2: Apply Basic Multiplication

First, calculate \((-5) imes (-5)\). Multiplying two negative numbers results in a positive number: \[(-5) imes (-5) = 25.\]

Step 3: Multiply Again

Now, take the result from Step 2 and multiply by \(-5\) one more time:\[25 imes (-5) = -125.\]

Step 4: Write the Final Result

The value of \((-5)^3\) is \[-125.\]

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign (-). They are used to describe values below zero, like temperatures in a polar vortex or debts. Understanding how negative numbers interact with each other during operations like addition, subtraction, multiplication, and division is crucial.

• Adding a negative number is like subtracting a positive number. For example, 5 plus -3 results in 2.
• Subtracting a negative number is like adding a positive number, turning -3 - (-2) into -3 + 2, which equals -1.
• When multiplying two negative numbers, the negatives "cancel out," leading to a positive result. For instance, \((-5) \times (-5) = 25\).
• Dividing two negative numbers also results in a positive number.

These rules are handy while learning multiplication and evaluating expressions, ensuring consistent results as seen in calculating exponents like \((-5)^3\).

Basic Multiplication

Basic multiplication is one of the four fundamental operations in arithmetic, involving the scaling of numbers. It allows us to find out how many times one number is contained within a certain quantity of another number.
Here, multiplication is simply repeated addition. For example, multiplying 4 by 3 is the same as adding 4 three times: 4 + 4 + 4 = 12.
In the expression \((-5)^3\), basic multiplication is key. We multiply \(-5\) by itself three times:

• First, we calculate \((-5) \times (-5)\), resulting in 25 since two negatives make a positive.
• Next, we multiply the result \(25\) by \(-5\) once again, which becomes \(-125\) as a positive and a negative give a negative product.

These steps show the simplicity and importance of understanding basic multiplication skills.

Order of Operations

The order of operations is a set of rules used to ensure that mathematical expressions are evaluated in a systematic and correct way. Following this order avoids confusion and errors.
The common mnemonic "PEMDAS" helps recall the order:

• Parentheses - Solve expressions within parentheses first.
• Exponents - Then, calculate powers or square roots.
• Multiplication and Division - From left to right.
• Addition and Subtraction - From left to right.

When dealing with expressions like \((-5)^3\), the order directly impacts our approach. Exponents are evaluated before other operations, ensuring that we deal with the exponentiation entirely before any other calculation.

Evaluating Expressions

Evaluating an expression means finding its numerical value by executing all the operations it includes. It involves a systematic approach and adherence to the order of operations.
For the expression \((-5)^3\), evaluating involves understanding what the exponent tells us: multiply \(-5\) by itself three times. This process requires several steps.

• Identify the operation. We see the power 3 and understand \(-5\) must be multiplied by itself thrice.
• Use basic multiplication, performing \((-5) \times (-5)\) to get 25.
• Finally, multiply the result by \(-5\) to achieve \(-125\). This shows how expressions can be broken down and calculated efficiently.

Patience and practice help in mastering the evaluation of complex expressions smoothly and confidently.