Powers and Indices: Basic Operations

TL;DR

Powers (or indices) are a shorthand way to write repeated multiplication of the same number. You'll learn how to multiply and divide terms with powers, and how to handle powers of powers. Mastering these rules makes complex calculations much simpler.

1. The Mental Model

Think of powers as a compact way of counting how many times a number is multiplied by itself. It's like having a special counter that shows how many copies of the base number are in the multiplication.

2. The Core Material

Powers and indices (they mean the same thing!) describe how many times a base number is multiplied by itself. For example, in $2^3$, 2 is the base and 3 is the power or index. It means $2 \times 2 \times 2$.

Multiplying Powers with the Same Base

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When you multiply terms with the same base, you add their powers.
The rule is: $a^m \times a^n = a^{m+n}$

• Why it works: Let's say you have $2^3 \times 2^2$.
  $2^3$ is $2 \times 2 \times 2$.
  $2^2$ is $2 \times 2$.
  So, $(2 \times 2 \times 2) \times (2 \times 2)$ means you're multiplying 2 by itself 5 times in total, which is $2^5$. And $3+2=5$.

Dividing Powers with the Same Base

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When you divide terms with the same base, you subtract their powers.
The rule is: $a^m \div a^n = a^{m-n}$

• Why it works: Take $3^5 \div 3^2$.
  $3^5$ is $3 \times 3 \times 3 \times 3 \times 3$.
  $3^2$ is $3 \times 3$.
  So, $\frac{3 \times 3 \times 3 \times 3 \times 3}{3 \times 3}$. You can cancel out two '3's from the top and bottom, leaving $3 \times 3 \times 3$, which is $3^3$. And $5-2=3$.

Power of a Power

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When you raise a power to another power, you multiply the powers.
The rule is: $(a^m)^n = a^{m \times n}$

• Why it works: Consider $(4^2)^3$.
  $4^2$ is $4 \times 4$.
  So, $(4^2)^3$ means $(4 \times 4) \times (4 \times 4) \times (4 \times 4)$.
  This is $4 \times 4 \times 4 \times 4 \times 4 \times 4$, which is $4^6$. And $2 \times 3=6$.

Zero Power

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Any non-zero number raised to the power of zero is 1.
The rule is: $a^0 = 1$ (where $a \neq 0$)

• Why it works: Using the division rule, $a^m \div a^m = a^{m-m} = a^0$.
  We also know that any number divided by itself is 1 (as long as it's not zero). So, $a^m \div a^m = 1$. Therefore, $a^0 = 1$.

Here's a quick overview of these rules:

graph TD
    A["Start with a calculation"] --> B{"Same Base?"}
    B -- "Yes, and Multiplying" --> C["Add Powers: a^(m+n)"]
    B -- "Yes, and Dividing" --> D["Subtract Powers: a^(m-n)"]
    B -- "No, but it's a Power of a Power" --> E["Multiply Powers: a^(m*n)"]
    B -- "Yes, and Power is Zero" --> F["Result is 1 (if base is not 0)"]
    C --> G["End"]
    D --> G
    E --> G
    F --> G

3. Worked Example

Let's simplify the expression: $(x^4 \times x^2) \div x^3$

1. First, simplify the multiplication inside the parentheses:
   Using the rule $a^m \times a^n = a^{m+n}$:
   $x^4 \times x^2 = x^{4+2} = x^6$

2. Now, substitute this back into the expression:
   The expression becomes $x^6 \div x^3$

3. Finally, perform the division:
   Using the rule $a^m \div a^n = a^{m-n}$:
   $x^6 \div x^3 = x^{6-3} = x^3$

So, $(x^4 \times x^2) \div x^3 = x^3$.

4. Key Takeaways

• To multiply powers with the same base, you add the exponents: $a^m \times a^n = a^{m+n}$.
• To divide powers with the same base, you subtract the exponents: $a^m \div a^n = a^{m-n}$.
• To raise a power to another power, you multiply the exponents: $(a^m)^n = a^{m \times n}$.
• Any non-zero base raised to the power of zero is always 1: $a^0 = 1$.
• Remember that these rules only apply when the bases are the same (except for power of a power).

Common mistakes to avoid:
- Don't add powers when dividing (e.g., $x^6 \div x^3 \neq x^9$).
- Don't multiply powers when multiplying terms (e.g., $x^2 \times x^3 \neq x^6$).
- Don't apply these rules to terms with different bases (e.g., $2^3 \times 3^2$ cannot be simplified using these rules).
- Don't forget that a zero power results in 1, not 0.

5. Now Try It

Simplify the expression: $(y^7 \div y^3) \times (y^2)^3$.
What success looks like: You should get a single term with 'y' raised to a specific power, applying each rule correctly step-by-step.