Problem 2 - Precalculus: Functions and Graphs

Question

Express the number in the form \(a / b,\) where \(a\) and \(b\) are integers. $$(-3)^{3}$$

Step-by-Step Solution

Answer

-27/1

Step 1: Understand the expression

The expression \((-3)^{3}\)\ indicates that we need to multiply -3 by itself three times.

Step 2: Multiply the first two numbers

To simplify \((-3)^{3}\), first multiply the first two \(-3 imes -3 = 9\)\.

Step 3: Multiply the result by the last number

Take the result from Step 2 and multiply it by \(-3\):\(9 imes -3 = -27\)\.

Step 4: Express the result as a fraction

Since the result of \((-3)^{3}\)\ is -27, express it in the form \(a/b\). Here, \(a = -27\) and \(b = 1\). Thus, the number is expressed as \(-27/1\).

Key Concepts

Understanding Integers

Integers are a fundamental part of mathematics, consisting of whole numbers that are both positive and negative, including zero. They do not include fractions or decimals. In this context, integers represent counts or whole units. For example, -3 is an integer that indicates three units in the negative direction. This concept is crucial when performing arithmetic operations, such as exponentiation.

Integer properties include:

• They are closed under addition, subtraction, and multiplication, meaning if you add, subtract, or multiply integers, the result is always an integer.
• They have a distinct property of being either negative, positive, or zero.
• The absolute value of an integer is the distance from zero on the number line, ignoring the sign.

This understanding is essential when dealing with exponentiation. Raising an integer like -3 to a power involves multiple multiplications of the integer by itself.

Exploring Fractions

Fractions describe a part of a whole, represented by two integers: the numerator and the denominator. In our exercise, the fraction form of the integer -27 is \(-27/1\). This indicates that -27 is equivalent to dividing -27 by 1, highlighting the concept that any integer can be expressed as a fraction where the denominator is 1.

Understanding fractions involves knowing:

• The numerator is the number on top, indicating how many parts we have.
• The denominator is the number below, showing into how many parts the whole is divided.
• If the denominator is 1, the fraction represents a whole number, just like -27/1 represents -27.

This can be particularly useful in converting numbers for calculations in different mathematical contexts.

The Process of Multiplication

Multiplication is a key arithmetic operation involving two or more numbers to find their total when combined at a rate specified by one of the numbers. For example, multiplying -3 by itself ultimately helps in simplifying expressions like \((-3)^3\).

Key aspects of multiplication include:

• It is commutative, meaning \(a \times b = b \times a\).
• It is associative, which means \((a \times b) \times c = a \times (b \times c)\).
• The concept of repeated addition, where multiplying by a number signifies adding it multiple times; for instance, \(-3 \times 3\ = -3 + (-3) + (-3)\).

In exponentiation, like \((-3)^3\), multiplication ensures the base number is repeatedly multiplied by itself, producing a resultant integer.