Algebraic Roots & Indices

Algebraic roots and indices (also known as exponents or powers) are essential topics in mathematics, spanning multiple grade levels from middle school to high school and beyond. Mastering the laws of indices helps simplify expressions, solve equations efficiently, and understand more advanced concepts in algebra, calculus, and number theory.

In this set of notes, we will explore:

Basic definitions of indices and roots
The complete laws (or rules) of indices
Negative, fractional, and zero exponents
Combining and simplifying expressions with indices
Rationalizing denominators when roots are involved
Progressive examples, from easy to advanced, with detailed solutions

1. Basic Definitions

1.1 Indices (Exponents)

When we write a number or variable raised to a power, such as $$a^n,$$ the value n is called the index or exponent, and a is called the base. The exponent tells us how many times the base is multiplied by itself:

$$a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ times}}.$$

Examples:

$$3^2 = 3 \times 3 = 9.$$
$$5^3 = 5 \times 5 \times 5 = 125.$$
$$x^4 = x \times x \times x \times x.$$

1.2 Roots

A root of a number is essentially the inverse (opposite operation) of raising that number to a power. For example, the square root is the opposite of squaring, and the cube root is the opposite of cubing.

The square root of a non-negative number $$a$$ is written as $$\sqrt{a},$$ and the cube root is written as $$\sqrt[3]{a}.$$ In general, an nth root is written:

$$\sqrt[n]{a} = a^{\frac{1}{n}},$$

showing that taking an nth root is equivalent to raising the number to the power of $$\frac{1}{n}.$$

2. Complete Laws of Indices

The laws of indices (or exponent rules) allow us to manipulate expressions involving powers. They apply to both numerical bases and algebraic bases (variables). Here are the primary rules you need to know.

Product Rule: $$a^m \times a^n = a^{m+n}.$$
When multiplying expressions with the same base, add the exponents.
Quotient Rule: $$\frac{a^m}{a^n} = a^{m-n}, \quad a \neq 0.$$
When dividing expressions with the same base, subtract the exponents.
Power of a Power Rule: $$(a^m)^n = a^{mn}.$$
When an expression with an exponent is raised to another exponent, multiply the exponents.
Power of a Product Rule: $$(ab)^n = a^n \, b^n.$$
An exponent distributed over a product can be applied to each factor separately.
Power of a Quotient Rule: $$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}, \quad b \neq 0.$$
An exponent distributed over a fraction can be applied to the numerator and denominator separately.
Zero Exponent Rule: $$a^0 = 1, \quad a \neq 0.$$
Any non-zero number raised to the power 0 equals 1.
Negative Exponent Rule: $$a^{-n} = \frac{1}{a^n}, \quad a \neq 0.$$
A negative exponent indicates a reciprocal.
Fractional Exponents: $$a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m.$$
A fractional exponent represents roots: the denominator of the fraction indicates the root, while the numerator indicates the power.

3. Examples and Exercises

Let’s illustrate these laws with a range of examples, starting from easy and progressing to more advanced problems.

3.1 Basic (Easy) Examples

Example 1: Simplify $$3^2 \times 3^3.$$

Solution: Use the product rule $$a^m \times a^n = a^{m+n}.$$ We have the same base (3), so:

$$3^2 \times 3^3 = 3^{2+3} = 3^5 = 243.$$

Example 2: Simplify $$x^4 \times x.$$

Solution: Remember that $$x = x^1.$$ So

$$x^4 \times x^1 = x^{4+1} = x^5.$$

Example 3: Simplify $$7^0.$$

Solution: By definition, any nonzero base to the zero power is 1:

$$7^0 = 1.$$

3.2 Intermediate Examples

Example 4: Simplify $$\frac{a^6}{a^2}.$$

Solution: Use the quotient rule $$\frac{a^m}{a^n} = a^{m-n}.$$ We get:

$$\frac{a^6}{a^2} = a^{6-2} = a^4.$$

Example 5: Simplify $$(-2)^3.$$

Solution: The negative sign is inside parentheses, so we multiply -2 by itself 3 times:

$$(-2)^3 = (-2) \times (-2) \times (-2).$$

First multiplication: $$(-2) \times (-2) = 4.$$
Second multiplication: $$4 \times (-2) = -8.$$

Therefore, $$(-2)^3 = -8.$$

Example 6: Simplify $$(x^3)^4.$$

Solution: Use the rule $$(a^m)^n = a^{mn}.$$ Hence:

$$(x^3)^4 = x^{3 \times 4} = x^{12}.$$

Example 7: Simplify $$\left(\frac{2x}{3y}\right)^2.$$

Solution: Apply $$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}.$$ and the product rule for exponents:

$$\left(\frac{2x}{3y}\right)^2 = \frac{(2x)^2}{(3y)^2} = \frac{2^2 \cdot x^2}{3^2 \cdot y^2} = \frac{4 x^2}{9 y^2}.$$

3.3 Negative and Fractional Exponents

Example 8: Rewrite $$x^{-3}$$ without negative exponents.

Solution: By definition:

$$x^{-3} = \frac{1}{x^3}.$$

Example 9: Simplify $$\frac{1}{x^{-2}}.$$

Solution: First rewrite the denominator:

$$x^{-2} = \frac{1}{x^2}.$$

So $$\frac{1}{x^{-2}} = \frac{1}{\frac{1}{x^2}} = x^2.$$

Example 10: Evaluate $$16^{\frac{1}{2}}.$$ (Square root form)

Solution: A fractional exponent with denominator 2 indicates the square root:

$$16^{\frac{1}{2}} = \sqrt{16} = 4.$$

Example 11: Simplify $$8^{\frac{2}{3}}.$$

Solution: Rewrite using roots:

$$8^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2.$$

Since $$\sqrt[3]{8} = 2,$$ we get

$$\left(\sqrt[3]{8}\right)^2 = 2^2 = 4.$$

3.4 Advanced Examples (Combining Multiple Laws)

Example 12: Simplify $$\frac{x^3 \, y^{-2}}{x^5 \, y^4}.$$

Solution Steps:

Combine the exponents of x:
$$x^3 \div x^5 = x^{3-5} = x^{-2}.$$
Combine the exponents of y:
$$y^{-2} \div y^4 = y^{-2-4} = y^{-6}.$$
So the expression becomes $$x^{-2} \, y^{-6} = \frac{1}{x^2} \times \frac{1}{y^6} = \frac{1}{x^2 \, y^6}.$$

Answer: $$\frac{1}{x^2 y^6}.$$

Example 13: Simplify $$\left(\frac{a^3 b^{-4}}{a^{-2} b^5}\right)^2.$$

Solution Sketch:

Simplify inside the parentheses first:
- Combine a: $$a^3 \div a^{-2} = a^{3 - (-2)} = a^{3+2} = a^5.$$
- Combine b: $$b^{-4} \div b^5 = b^{-4-5} = b^{-9}.$$
So inside parentheses you have $$a^5 b^{-9}.$$
Now raise to the power 2:
$$\left(a^5 b^{-9}\right)^2 = a^{5 \times 2} \, b^{-9 \times 2} = a^{10} b^{-18}.$$
Rewrite negative exponents:
$$a^{10} b^{-18} = a^{10} \frac{1}{b^{18}} = \frac{a^{10}}{b^{18}}.$$

Answer: $$\frac{a^{10}}{b^{18}}.$$

Example 14: Simplify $$\left(\sqrt{x^3 y}\right)^2 \div x \, y^{\frac{3}{2}}.$$

Solution Approach:

Rewrite the square root in exponent form: $$\sqrt{x^3 y} = (x^3 y)^{\frac{1}{2}} = x^{\frac{3}{2}} \, y^{\frac{1}{2}}.$$
Raise it to the power 2:
$$\left(x^{\frac{3}{2}} \, y^{\frac{1}{2}}\right)^2 = x^{\frac{3}{2} \times 2} \, y^{\frac{1}{2} \times 2} = x^3 \, y^1 = x^3 y.$$
Now we have $$\frac{x^3 y}{x \, y^{\frac{3}{2}}} = x^{3-1} \, y^{1 - \frac{3}{2}} = x^2 \, y^{-\frac{1}{2}}.$$
Rewrite $$y^{-\frac{1}{2}} = \frac{1}{y^{\frac{1}{2}}} = \frac{1}{\sqrt{y}}.$$

Answer: $$x^2 \, y^{-\frac{1}{2}} = \frac{x^2}{\sqrt{y}}.$$

4. Roots and Rationalization

In algebra, especially when dealing with surds (irrational roots), “rationalizing the denominator” is a key skill. This means rewriting an expression so that there are no irrational (root) expressions in the denominator.

4.1 Rationalizing Denominators

Example 15 (Easy): Rationalize $$\frac{3}{\sqrt{2}}.$$

Solution: Multiply numerator and denominator by $$\sqrt{2}$$:

$$\frac{3}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{2}.$$

Example 16 (Moderate): Rationalize $$\frac{4}{2 + \sqrt{3}}.$$

Solution: Multiply top and bottom by the conjugate of the denominator, which is $$2 - \sqrt{3}.$$

$$\frac{4}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{4(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})}.$$

Recall that $$(a + b)(a - b) = a^2 - b^2.$$ So here:

Denominator = $$(2)^2 - (\sqrt{3})^2 = 4 - 3 = 1.$$

Therefore:

$$\frac{4(2 - \sqrt{3})}{1} = 8 - 4\sqrt{3}.$$

Answer: $$8 - 4\sqrt{3}.$$

5. Further Practice Exercises

Test your understanding with these practice problems. Attempt to solve them before looking at any hints or solutions.

5.1 Easy Exercises

Simplify $$5^2 \times 5^1.$$
Simplify $$x^0.$$
Write $$\sqrt{25}$$ as a power of 25.
Express $$\frac{1}{x^3}$$ as a negative exponent.

5.2 Moderate Exercises

Simplify $$x^3 \times x^5 \div x^6.$$
Evaluate $$81^{\frac{1}{4}}.$$ (Hint: $$81 = 3^4$$)
Rationalize $$\frac{2}{\sqrt{5}}.$$
Rewrite $$\left(\frac{2}{3}\right)^{-2}$$ in simplest form.

5.3 Advanced Exercises

Simplify $$\frac{x^3 y^{-1}}{x^4 y^{-3}}.$$
Simplify $$\left(\frac{a^2 b^{-3}}{a^{-4} b^2}\right)^3.$$
Rationalize $$\frac{1}{\sqrt{2} - \sqrt{3}}.$$ (Hint: Multiply top and bottom by $$\sqrt{2} + \sqrt{3}.$$)
Rewrite $$64^{\frac{2}{3}} \div 4^{\frac{1}{2}}$$ in simplest form.

6. Common Mistakes & Reminders

Forgetting to add exponents when multiplying with the same base. Always ensure you have the same base to apply the product rule, or you might need to factor or rewrite bases to match.
Mishandling negative exponents. A negative exponent indicates a reciprocal; never leave a final expression with a negative exponent unless instructed otherwise.
Ignoring parentheses on negative bases. $(-2)^3 \neq -2^3 \, (\text{the latter is } -(2^3) = -8). Pay close attention to parentheses to determine if the negative sign is raised to the power.
Missing the idea of fractional exponents as roots. $$a^{\frac{m}{n}} = \sqrt[n]{a^m}.$
Improper rationalizing methods. When the denominator is a sum or difference of square roots, use the conjugate technique: multiply top and bottom by $$a - b$$ if your denominator is $$a + b.$$

7. Conclusion

Understanding how to work with indices (exponents) and roots is a crucial skill in algebra and beyond. These laws are the building blocks for simplifying expressions, solving exponential equations, and manipulating surds. The step-by-step examples above should help you grasp the concepts from a fundamental to an advanced level.

With ample practice, you’ll find that these rules become second nature, boosting your confidence and accuracy in various mathematical contexts from basic school-level problems to more complex high-school and even college-level algebra.