Chapter 2: Algebra

This chapter revisits some fundamental algebraic rules involving fractions, exponents, polynomials, and the use of parentheses. A firm grasp of these ideas is essential, since many common mistakes in computation and symbolic manipulation arise from misunderstanding or misapplying these basic principles.

Algebraic expressions and their underlying rules appear across a wide range of mathematical contexts. Developing the ability to recognize, interpret, and manipulate such expressions is therefore a key skill—both for simplifying symbolic formulas and for solving more complex problems in later chapters.

Note

Unless otherwise specified, all constants, variables, and placeholders are assumed to belong to subsets of the real numbers (i.e., $R$ ).

In other words, when we perform standard operations such as addition, subtraction, multiplication, or division (except division by zero), the results remain within $R$ .

Basic Algebraic Properties

Before we explore more advanced algebraic concepts, it is useful to recall a few basic properties that govern addition and multiplication. These properties, i.e., the commutative, associative, and distributive laws, apply to all real numbers and allow us to manipulate expressions, regardless of how they are written or grouped.

The Commutative, Associative & Distributive Law

The commutative law states that the order of two elements does not affect the result:

$a + b = b + a and a \cdot b = b \cdot a$

The associative law states that the way elements are grouped does not affect the result:

$(a + b) + c = a + (b + c) and (a \cdot b) \cdot c = a \cdot (b \cdot c)$

The distributive law of multiplication over addition (and subtraction):

$a \cdot (b + c) = a \cdot b + a \cdot c$

This property allows us to distribute a factor across terms inside parentheses.

Example 1: Distributive Law

Consider the expression:

$- 2 \cdot (3 + 5)$

Using the distributive law, we multiply $- 2$ by each term inside the parentheses:

$- 2 \cdot 3 + (- 2) \cdot 5 = - 6 - 10 = - 16$

The result is the same as first adding the terms inside the parentheses and then multiplying:

$- 2 \cdot (8) = - 16$

This confirms that the distributive and associative properties are consistent, i.e., the order in which we group or distribute the factors does not change the result.

Example 2: Distributive Law

Consider the expression:

$- (4 - 7)$

Here, the negative sign in front of the parentheses can be interpreted as multiplying by $- 1$ :

$- (4 - 7) = (- 1) \cdot (4 - 7)$

Applying the distributive law, we multiply $- 1$ by each term inside the parentheses:

$(- 1) \cdot 4 + (- 1) \cdot (- 7) = - 4 + 7 = 3$

This shows that placing a negative sign in front of parentheses changes the sign of each term inside.

Fractions

Fractions represent parts of a whole and are especially useful when dealing with proportions, ratios, and percentages. A fraction consists of two parts:

A numerator (top number): represents how many parts we have.
A denominator (bottom number): represents how many equal parts make up the whole.

In symbolic form, a fraction is written as a ratio of two integers:

$\frac{a}{b}, where a, b \in Z, b \neq = 0$

The set of all such numbers is called the rational numbers and denoted by $Q$ , and it forms a subset of the real numbers:

$Q \subset R$

Warning: A Common Mistake When Adding Fractions

Adding fractions is not done by simply adding the numerators and denominators:

$\frac{a}{b} + \frac{c}{d} \neq = \frac{a + c}{b + d}$

For example:
$\frac{1}{3} + \frac{1}{3} = \frac{2}{3}$ but $\frac{1 + 1}{3 + 3} = \frac{2}{6} = \frac{1}{3}$ which is incorrect for addition. Always use a common denominator as explained below.

Rule: Addition of Fractions

To add or subtract fractions, the denominators must be the same. Once a common denominator is found, we can add (or subtract) the numerators and keep the denominator unchanged.

If the denominators are already the same: $\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$

If they are different, multiply each numerator by the other fraction’s denominator to obtain a common base: $\frac{a}{c} + \frac{b}{d} = \frac{a \cdot d + b \cdot c}{c \cdot d}$

Examples: Adding and Subtracting Fractions

Evaluate the following expression (fractions with the same denominator): $\frac{2}{5} + \frac{1}{5} = \frac{2 + 1}{5} = \frac{3}{5}$
Evaluate the following expression (fractions with different denominators): $\frac{1}{2} + \frac{1}{3} = \frac{1 \cdot 3 + 1 \cdot 2}{2 \cdot 3} = \frac{3 + 2}{6} = \frac{5}{6}$
Evaluate the following expression (subtracting two fractions): $\frac{5}{6} - \frac{1}{3} = \frac{5 \cdot 3 - 1 \cdot 6}{6 \cdot 3} = \frac{15 - 6}{18} = \frac{9}{18} = \frac{1}{2}$

Rule: Multiplication of Fractions

Multiplication of fractions is straightforward: multiply the numerators together and the denominators together.

$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$

Examples: Multiplying Fractions

Evaluate the following expression (multiply two fractions directly): $\frac{2}{3} \cdot \frac{3}{4} = \frac{2 \cdot 3}{3 \cdot 4} = \frac{6}{12} = \frac{1}{2} = 0.5$ Multiplying straight across gives $\frac{6}{12}$ , which simplifies to $\frac{1}{2}$ .
Evaluate the following expression (simplify before multiplying): $\frac{5}{6} \cdot \frac{2}{5} = \frac{5 \cdot 2}{6 \cdot 5} = \frac{10}{30} = \frac{1}{3} \approx 0.33$ Since $5$ appears in both numerator and denominator, it can be simplified before or after multiplication.
Evaluate the following expression (multiply a whole number by a fraction): $3 \cdot \frac{2}{5} = \frac{3}{1} \cdot \frac{2}{5} = \frac{3 \cdot 2}{1 \cdot 5} = \frac{6}{5} = 1.20$ Whole numbers can be treated as fractions with denominator $1$ , making the same rule apply.

Rule: Division of Fractions

To divide one fraction by another, multiply the first fraction by the reciprocal (or multiplicative inverse) of the second fraction.

$\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \cdot \frac{d}{c} = \frac{a \cdot d}{b \cdot c}$

Examples: Dividing Fractions

Evaluate the following expression (divide one fraction by another): $\frac{\frac{3}{4}}{\frac{2}{5}} = \frac{3}{4} \cdot \frac{5}{2} = \frac{3 \cdot 5}{4 \cdot 2} = \frac{15}{8} = 1.875$ The reciprocal of $\frac{2}{5}$ is $\frac{5}{2}$ ; multiplying gives $\frac{15}{8}$ .
Evaluate the following expression (divide by a smaller fraction): $\frac{\frac{5}{6}}{\frac{1}{2}} = \frac{5}{6} \cdot \frac{2}{1} = \frac{5 \cdot 2}{6 \cdot 1} = \frac{10}{6} = \frac{5}{3} \approx 1.66$ Dividing by a smaller fraction increases the result, since $\frac{1}{2}$ fits multiple times into $\frac{5}{6}$ .
Evaluate the following expression (divide a fraction by a whole number): $\frac{\frac{7}{9}}{7} = \frac{\frac{7}{9}}{\frac{7}{1}} = \frac{7}{9} \cdot \frac{1}{7} = \frac{7 \cdot 1}{9 \cdot 7} = \frac{7}{63} = \frac{1}{9} \approx 0.11$ Note here that the whole number $7$ can be written as $\frac{7}{1}$ , and its reciprocal is $\frac{1}{7}$ .

Note

The rational numbers $Q$ are closed under addition, subtraction, multiplication, and division (except division by zero). This means performing these operations on fractions always produces another rational number.

Exponents

Exponents indicate how many times a base number is multiplied by itself. For example:

$1^{2} = 1 \cdot 1 or 4^{3} = 4 \cdot 4 \cdot 4$

In these expressions, the base ( $1$ and $4$ , respectively) tells us what to multiply, while the exponent ( $2$ and $3$ , respectively) tells us how many times to multiply it.

Exponents essentially provide a compact way to represent repeated multiplication and follow a consistent set of algebraic rules, which we go into details with below.

Rule: Power of Zero

For any nonzero base $a$ , raising it to the power of zero equals 1:

$a^{0} = 1$

The reason $a$ must be nonzero is to avoid ambiguity. Depending on how we reason about it, we can arrive at two conflicting interpretations. From one perspective, since $0^{x} = 0$ for any positive $x$ , it might seem natural to conclude that $0^{0} = 0$ . From another, because $a^{0} = 1$ for any positive $a$ , one could instead argue that $0^{0} = 1$ . These two lines of reasoning contradict each other, so $0^{0}$ is left undefined to avoid inconsistency.

Rule: Product of Powers

When multiplying powers that share the same (nonzero) base ( $a$ ), we add their exponents ( $n$ and $m$ ):

$a^{n} \cdot a^{m} = a^{n + m}$

This rule follows from the idea that each exponent represents repeated multiplication of the same base, and combining them extends that repetition into just a single product.

Examples: Product of Powers

In the following we apply the Product of Powers rule to see how it works. That is, we evaluate the following expression: $2^{4} \cdot 2^{2} = 4 + 2 = 6 factors in total (2 \cdot 2 \cdot 2 \cdot 2) \cdot (2 \cdot 2) = 2^{6} = 64$

Here, each exponent counts how many times the base 2 appears as a factor. Combining both terms gives $4 + 2 = 6$ factors of 2 in total.

Rule: Power of a Power

When raising an exponential term ( $a^{n}$ ) to another power ( $m$ ), we simply multiply the exponents:

$(a^{n})^{m} = a^{n \cdot m}$

This rule reflects that each copy of $a^{n}$ contributes $n$ factors of $a$ , and there are $m$ such copies in total, giving us $n \cdot m$ factors altogether.

Examples: Power of a Power

In the following we apply the Power of a Power rule to see how it works. That is, we evaluate the following expression:

$(5^{3})^{2} = 5^{3} \cdot 5^{3} = 3 \cdot 2 = 3 + 3 = 6 factors in total (5 \cdot 5 \cdot 5) \cdot (5 \cdot 5 \cdot 5) = 5^{3 \cdot 2} = 5^{6} = 15, 625$

Here, the inner exponent ( $3$ ) tells us there are three factors of 5 in each group, and the outer exponent ( $2$ ) tells us there are two such groups. Altogether, we get $3 \times 2 = 6$ factors of 5.

Rule: Negative Exponent

When a base ( $a$ ) is raised to a negative exponent ( $- n$ ), the result is the reciprocal of the base ( $\frac{1}{a}$ ) raised to the corresponding positive exponent ( $n$ ):

$a^{- n} = \frac{a ^{- n}}{1} = \frac{1}{a ^{n}}, a \neq = 0$

This rule essentially expresses that a negative exponent "flips" the base, moving it from the numerator to the denominator.

Examples: Negative Exponent

Let us apply the Negative Exponent rule to see how it works.

First let us evaluate an expresseion when the base is positive:

$2^{- 3} = \frac{1}{2 ^{3}} = \frac{1}{2 \cdot 2 \cdot 2} = \frac{1}{8}$

Next, we evaluate an expression when the base is negative:

$(- 5)^{- 2} = \frac{1}{( - 5 ) ^{2}} = \frac{1}{- 5 \cdot - 5} = \frac{1}{25}$

Rule: Quotient of Powers

When dividing powers that share the same base ( $a$ ), then we subtract the exponents ( $n$ and $m$ ).

This rule follows directly from the Product of Powers and Negative Exponent rules, i.e., division is simply multiplication by the reciprocal: $\frac{a ^{n}}{a ^{m}} = a^{n} \cdot \frac{1}{a ^{m}} = a^{n} \cdot a^{- m} = a^{n - m} (Use: Negtative Exponent Rule) (Use: Product Rule)$

This rules apply when $a \neq = 0$ , since division by zero is undefined.

Examples: Quotient of Powers

Let us apply the Quotient of Powers rule to see how it works directly:

$\frac{3 ^{5}}{3 ^{2}} = 3^{5 - 2} = 3^{3} = 3 \cdot 3 \cdot 3 = 27$

We can also expand the numerator and denominator to illustrate how factors cancel out:

$\frac{3 ^{5}}{3 ^{2}} = \frac{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}{3 \cdot 3} = 3 \cdot 3 \cdot 3 = 3^{5 - 2} = 3^{3} = 27$

Here, the two factors of 3 in the denominator remove two factors from the numerator, leaving $5 - 2 = 3$ factors in total.

Rule: Fractional Exponents (Roots as Powers)

Roots can be expressed as fractional exponents. In general, the $r$ -th root of $a$ can be written as follows:

$a^{\frac{1}{r}} = r a$

This allows us to apply exponent rules even when working with roots, as roots are simply another form of exponentiation.

Examples: Fractional Exponents (Roots as Powers)

Let us apply the rule by expressing roots as fractional exponents:

The square root of a number: $a = a^{\frac{1}{2}}$
The cube root of a number: $3 a = a^{\frac{1}{3}}$
The fourth root of a power: $4 a^{3} = a^{\frac{3}{4}}$

Here, the denominator of the exponent corresponds to the root, while the numerator corresponds to the power.

Rule: Product of Roots

The root of a product is equal to the product of the roots (for non-negative $a$ and $b$ ):

$(ab)^{\frac{1}{r}} = a^{\frac{1}{r}} \cdot b^{\frac{1}{r}}, a, b \geq 0$

For square roots ( $r = 2$ ), this simplifies to:

$ab = a \cdot b$

This property holds only for non-negative real numbers, since roots of negative numbers are not real (they care complex numbers).

Examples: Product of Roots

Let us apply the Product of Roots rule to simplify a root expression:

$1 2^{\frac{1}{2}} = (4 \cdot 3)^{\frac{1}{2}} = 4^{\frac{1}{2}} \cdot 3^{\frac{1}{2}} = 4 \cdot 3 = 2 \cdot 3 (Using the Product of Roots Rule) (Since a^{\frac{1}{2}} = a)$

Here, expressing $12$ as $4 \cdot 3$ allows us to separate the square root into two simpler factors,
making the simplification straightforward.

Warning:

All these exponent rules apply for any real exponent, not just integers.

However, it is generally not possible to simplify expressions such as $2^{3} \cdot 3^{5}$ when the bases are different and unrelated by a common factor.

Likewise, exponent rules do not apply to addition or subtraction, so expressions like $a^{n} + b^{m}$ cannot be simplified using these rules.

Algebraic Identities

When working with algebraic expressions, we often encounter recurring patterns that make calculations simpler. The commutative, associative, and distributive laws, together with the rules of exponents, provide the foundation for manipulating and simplifying such expressions.

In particular, by applying the distributive law repeatedly, and interpreting exponents like $(a + b)^{2}$ or $(a + b)^{3}$ as repeated multiplication, we can derive a number of useful algebraic identities. These identities, summarized in the table below, describe common patterns that occur when expanding or factoring expressions and offer compact formulas that are helpful in later algebraic work.

Table 1. Common algebraic identities derived from the distributive law.

Name	Expression	Factored Form	Expanded Form
Square of a Sum	$(a + b)^{2}$	$(a + b) (a + b)$	$a^{2} + 2 ab + b^{2}$
Square of a Difference	$(a - b)^{2}$	$(a - b) (a - b)$	$a^{2} - 2 ab + b^{2}$
Difference of Squares	$a^{2} - b^{2}$	$(a - b) (a + b)$	$a^{2} - b^{2}$

Mathematics Brush-up for Data Science

Chapter 2: Algebra

Basic Algebraic Properties

Fractions

Exponents

Algebraic Identities

Chapter Exercises

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Mathematics Brush-up for Data Science

Chapter 2: Algebra

Basic Algebraic Properties

Fractions

Exponents

Algebraic Identities

Chapter Exercises