Problem 2 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-10: Express the number in the form \(a / b\), where \(a\) and \(b\) are integers. $$ (-3)^{3} $$

Step-by-Step Solution

Answer

The number \((-3)^{3}\) is expressed as \(-27/1\).

Step 1: Understand the Exercise

The exercise asks us to express the number \((-3)^{3}\) in the form \(a/b\), where \(a\) and \(b\) are integers. This involves simplifying the expression \((-3)^{3}\).

Step 2: Calculate \((-3)^3\)

Calculate the cube of \(-3\): \((-3) imes (-3) imes (-3)\). Start by computing \((-3) imes (-3)\), which gives \(9\). Then multiply that result by \(-3\) to get \(-27\).

Step 3: Express as Fraction

Once you have \(-27\), write the number as a fraction with \(1\) as the denominator: \(-27/1\). This is in the form \(a/b\), where \(a = -27\) and \(b = 1\).

Step 4: Simplification

Observe that the fraction \(-27/1\) is already in its simplest form because the denominator is \(1\), which represents an integer \(-27\) itself.

Key Concepts

Cube of a Number

When we talk about the cube of a number, we refer to the process of multiplying a number by itself three times. This is an important concept in mathematics, particularly when dealing with powers and exponents. For instance, if we have a number like - \((-3)^3\), it simply means we multiply \((-3) \times (-3) \times (-3)\).The calculation starts with multiplying the first two - \(-3 \times -3 = 9\). This is because the product of two negative numbers is positive. Now, multiply the result by the third number:- \(9 \times (-3)\).Here, multiplying a positive by a negative results in a negative, giving us \(-27\). Understanding this multiplication helps simplify expressions involving cubes.

Fraction Representation

Fraction representation is an easy way to express whole numbers or results of operations as fractions. Even when a number can stand alone, putting it into this form can be beneficial, particularly when adhering to instructions, like in exercises that request results as fractions. In our example,- \(-27/1\), we write this fraction to express the number \(-27\) in the form of \(a/b\).Here, - \(a\) is \(-27\), and- \(b\) is \(1\),showing that any integer \(n\) can be expressed as \(n/1\).This method highlights the versatility of fractions in mathematics, showing how every integer is essentially a fraction with the denominator of one. Keeping this in mind is crucial for solving problems requiring fractional answers.

Simplification of Expressions

Simplification of expressions involves reducing an expression or fraction to its most basic or easiest form to understand. Simplifying can often make calculations much more straightforward. In our exercise, we simplified \(-27\) to- \(-27/1\).This fraction shows that \(-27\) is already in its simplest form because having a denominator of - \(1\)implies that it's just the number itself.A fraction is considered simplest when the numerator and denominator have no common factors other than - \(1\).And since \(-27\) can be written as \(-27/1\), there are no further reductions possible. Recognizing such expressions in their simplest form is a key mathematical skill that helps ensure solutions are presented clearly and correctly.