Algebraic Fractions & Laws of Indices

Algebraic fractions and indices are simplification tools. They help you rewrite complicated expressions into cleaner forms without changing their value.

In the CSEC exam, these skills often appear inside larger algebra questions rather than as a topic by itself. You may need to simplify before solving, factor before cancelling, or apply index laws before substituting. Each line should preserve equality, so explain what rule you are using when the step is not obvious.

Simplifying Algebraic Fractions¶

The safest way to simplify an algebraic fraction is to factor first, then cancel common factors. Cancelling terms that are being added is a serious algebra mistake.

Algebraic fractions work exactly like regular fractions — but with variables instead of just numbers.

The Key Principle: You can cancel something from top and bottom if it appears in BOTH places.

Rule 1: Cancel Common Variables¶

If the same variable appears on top and bottom, you can cross them out.

Rule 2: Factor the Numerator, Then Cancel¶

When the numerator is more complex, factor it first. Then look for common factors with the denominator.

---

Adding and Subtracting Algebraic Fractions¶

Addition and subtraction depend on matching denominators because the pieces must be the same size before they can be combined.

Just like regular fractions, you need a common denominator to add or subtract.

Case 1: Same Denominator (Easy)¶

If the denominators match, just combine the numerators.

Case 2: Different Denominators (Harder)¶

Find the Least Common Denominator (LCD) — the smallest expression both denominators divide into.

Finding LCD Strategy¶

---

Part 5: Laws of Indices (Exponents) — Powers and Their Rules¶

What Are Indices?¶

Indices are shorthand for repeated multiplication. They are not decoration; they control how many times the base is used as a factor.

Index (or exponent) is the small number that says "multiply this number by itself how many times."

$x^3 = x \times x \times x \text{ (three times)}$ $2^4 = 2 \times 2 \times 2 \times 2 = 16 \text{ (four times)}$

The index tells you how many times to use the base.

---

Rule 1: Multiply — Add the Indices¶

Use this rule only when the bases are the same. $x^2 \times x^3$ can be combined, but $x^2 \times y^3$ cannot become one power because the bases are different.

When you multiply powers with the SAME base, ADD the exponents:

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

Why? Because: $x^2 \times x^3 = (x \times x) \times (x \times x \times x) = x \times x \times x \times x \times x = x^5$

You're multiplying 2 of them times 3 of them = 5 of them total.

---

Rule 2: Divide — Subtract the Indices¶

Division removes repeated factors. Subtracting the exponents counts how many copies of the base remain after cancellation.

When you divide powers with the SAME base, SUBTRACT the exponents:

$\frac{a^m}{a^n} = a^{m-n}$

Why? Because: $\frac{x^5}{x^2} = \frac{x \times x \times x \times x \times x}{x \times x} = x \times x \times x = x^3$

Cancel two of them, and you have three left. $5 - 2 = 3$.

---

Rule 3: Power of a Power — Multiply the Indices¶

A power of a power means repeated groups of repeated factors. Multiplying the indices counts the total number of factors.

When you have a power raised to another power, MULTIPLY the exponents:

$(a^m)^n = a^{mn}$

Why? Because: $(x^2)^3 = x^2 \times x^2 \times x^2 = (x \times x) \times (x \times x) \times (x \times x) = x^6$

You have 3 groups of $x^2$, so $2 \times 3 = 6$ x's total.

---

Rule 4: Product Rule — Apply Power to Each Factor¶

When a whole product is raised to a power, every factor inside the brackets is repeated. This includes numerical coefficients, not only variables.

When you raise a product to a power, apply the power to EACH part:

$(ab)^n = a^n b^n$

Why? Because: $(xy)^3 = (xy) \times (xy) \times (xy) = (x \times x \times x) \times (y \times y \times y) = x^3 y^3$

---

Rule 5: Zero Power — Always Equals 1¶

The zero power rule often surprises students because it does not mean "nothing". It comes from a pattern of division where equal powers cancel completely.

Any number (except 0) raised to the power 0 equals 1:

$a^0 = 1$

Why? Think about the divide rule: $\frac{a^n}{a^n} = 1 \text{ (something divided by itself)}$

But also: $\frac{a^n}{a^n} = a^{n-n} = a^0$

So $a^0 = 1$.

---

Rule 6: Negative Powers — Flip to a Fraction¶

A negative exponent shows position, not negativity. It moves the factor to the other side of a fraction bar and makes the exponent positive.

A negative power means "put it in a fraction":

$a^{-n} = \frac{1}{a^n}$

Why? Using the divide rule: $\frac{a^0}{a^n} = a^{0-n} = a^{-n}$

But $a^0 = 1$, so: $a^{-n} = \frac{1}{a^n}$

---

All Six Rules Summary¶