What is the difference between $x^a \cdot x^b$ and $(x^a)^b$?

In $x^a \cdot x^b$, you are multiplying two separate powers with the same base, so you add the indices ($x^{a+b}$). In $(x^a)^b$, you are raising a power to another power, which means you multiply the indices ($x^{ab}$).

How does the treatment of a negative sign differ between $(-3)^2$ and $-3^2$?

In $(-3)^2$, the base is $-3$, so the negative is squared, resulting in $+9$. In $-3^2$, the base is only $3$, so the square is calculated first ($9$) and then the negative is applied, resulting in $-9$.

When should you use the reciprocal instead of a negative value for an index?

A negative index like $x^{-n}$ always indicates a reciprocal ($\frac{1}{x^n}$), never a negative numerical result. You use the reciprocal whenever the index is negative to move the term across the fraction bar.

What is the common error when simplifying $2^3 \times 2^4$?

A common mistake is multiplying the bases to get $4^7$. The correct approach is to keep the base the same and only add the indices, resulting in $2^7$.

Why is it incorrect to say $(x + y)^2 = x^2 + y^2$?

Indices only distribute over multiplication and division, not addition or subtraction. Squaring a binomial requires expansion (FOIL), which includes a middle term ($2xy$).

What happens if you assume $a^0 = 0$ in a calculation?

Assuming $a^0 = 0$ will lead to incorrect products in algebraic expressions. By definition, any non-zero number to the power of zero is 1, which preserves the identity of other factors in a product.

Define the 'Base' and the 'Index' in the expression $y^5$.

The base is $y$, which is the number or variable being repeatedly multiplied. The index is $5$, which tells us that $y$ appears as a factor five times ($y \cdot y \cdot y \cdot y \cdot y$).

What does a fractional index $a^{\frac{1}{n}}$ represent?

A fractional index with a numerator of 1 represents the $n$-th root of the base. For example, $a^{\frac{1}{2}}$ is the square root of $a$, and $a^{\frac{1}{3}}$ is the cube root of $a$.

State the Power of a Product Law.

The Power of a Product Law states that $(ab)^n = a^n b^n$. This means every factor inside the parentheses is raised to the power outside the parentheses.

What is the formula for a negative index $x^{-n}$?

The formula is $x^{-n} = \frac{1}{x^n}$. This rule allows us to convert negative exponents into positive exponents by taking the reciprocal of the base.

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2. Algebra & Functions

Indices

Summary

Indices, also known as exponents or powers, provide a mathematical shorthand for representing repeated multiplication of the same factor. The study of indices involves mastering a set of fundamental laws that govern how these powers interact during algebraic operations such as multiplication, division, and exponentiation.

1. Definition & Core Concepts

Base and Index: In the expression $a^n$ , the base ( $a$ ) is the number being multiplied, and the index ( $n$ ) indicates how many times the base is used as a factor. This notation is essential for simplifying long strings of multiplication into a compact, manageable form.
Repeated Multiplication: The expression $a^n$ is defined as $a \times a \times a \dots$ for $n$ factors. Understanding this physical representation helps in visualizing why certain laws, such as the addition of indices during multiplication, actually work.
Terminology: The terms exponent, power, and index are often used interchangeably in mathematics to describe the superscript value that dictates the repeated multiplication.

Diagram showing the components of an index: the base 'a' and the index 'n', with the definition of repeated multiplication.

2. Fundamental Laws of Indices

3. Zero, Negative, and Fractional Indices

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions