Problem 16 - Elementary Algebra

Question

For the following problems, write each of the quantities using exponential notation. (-3) cubed

Step-by-Step Solution

Answer

Answer: (-3)^3=-27

Step 1: Write the given quantity in exponential notation

We are given (-3) cubed. We can write this as (-3)^{3}.

Step 2: Simplify the given expression

To simplify (-3)^{3}, we multiply -3 by itself three times: \[-3\cdot -3\cdot -3\]. Since a negative number multiplied by another negative number yields a positive number, and a positive number multiplied by a negative number yields a negative number, this expression simplifies to: \[(-3)^{3}= -27.\] That's the final answer: \[(-3)^{3}=-27.\]

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is fundamental to grasping the basic concepts of algebra. An algebraic expression is a collection of numbers, variables (letters that represent unknown values), and operations (such as addition, subtraction, multiplication, and division). For instance, the expression \( (-3)^3 \) is an algebraic expression where \( -3 \) is the base and \( 3 \) is the exponent, suggesting we multiply \( -3 \) by itself two more times.

When dealing with algebraic expressions, it's essential to follow the correct order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Following this rule, we calculate exponents before multiplication or division. This systematic approach helps to unravel more complex algebraic expressions and ensures that you reach the correct solution.

Negative Numbers

Negative numbers are values less than zero, represented with a minus sign (–). They play a crucial role in the number system, allowing us to describe quantities that are lacking or below a defined level, such as temperature below freezing or a loss in profit. In the context of exponential notation as seen with \( (-3)^3 \), negative numbers behave according to specific rules when raised to powers.

A negative number raised to an odd power, as in our example, results in a negative number (\[ (-3)^3 = -27 \]). Conversely, if a negative number is raised to an even power, the result is positive because the product of two negative numbers is positive. Understanding how negative numbers operate under different scenarios is crucial to mastering the concepts of algebra.

Powers and Exponents

Powers and exponents express how many times a number, known as the base, is multiplied by itself. The exponent, located as a superscript to the right of the base (for example, \(3^2\)), dictates the number of times the base is used as a factor. Exponential notation is a concise way to represent repeated multiplication, and it's particularly useful in algebra for simplifying expressions.

Let's dissect the exponential notation \( (-3)^3 \). Here, \( -3 \) is the base and \( 3 \) is the exponent. To simplify, we multiply \( -3 \) by itself three times: \[ -3 \times -3 \times -3 = -27 \]. It is important to remember that the base \( -3 \) must be used as a whole, which means including its negative sign in each multiplication. This is a common area where mistakes can occur, so it's vital to be mindful of the base and its sign when simplifying exponential expressions.