Indices and Surds

TL;DR

Indices (also called powers) are a neat way to show repeated multiplication, and understanding their rules helps you simplify complex expressions. Surds are just square roots of non-square numbers, and you'll learn how to simplify and work with them. Both topics are about simplifying expressions to make them easier to handle.

1. The Mental Model

Think of indices as a shorthand for multiplying a number by itself many times, like 2^5 means 2 x 2 x 2 x 2 x 2. Surds are like "unsolvable" square roots, such as √7, which can't be written as a neat whole number or fraction, but you can often simplify parts of them.

2. The Core Material

What are Indices?

An index (plural: indices) tells you how many times to multiply a number by itself. For example, in 3^4, 3 is the base and 4 is the index (or power). It means 3 x 3 x 3 x 3.

Key Rules of Indices

1. Multiplication Rule: When you multiply numbers with the same base, you add the powers.
   a^m * a^n = a^(m+n)
   Example: 2^3 * 2^4 = 2^(3+4) = 2^7

2. Division Rule: When you divide numbers with the same base, you subtract the powers.
   a^m / a^n = a^(m-n)
   Example: 5^6 / 5^2 = 5^(6-2) = 5^4

3. Power of a Power Rule: When you raise a power to another power, you multiply the powers.
   (a^m)^n = a^(m*n)
   Example: (3^2)^3 = 3^(2*3) = 3^6

4. Zero Power Rule: Any number (except 0) raised to the power of zero is 1.
   a^0 = 1 (where a ≠ 0)
   Example: 7^0 = 1

5. Negative Power Rule: A negative power means the reciprocal of the base raised to the positive power.
   a^(-n) = 1 / a^n
   Example: 4^(-2) = 1 / 4^2 = 1 / 16

6. Fractional Power Rule: A fractional power 1/n means the nth root. A fractional power m/n means the nth root, then raised to the power m.
   a^(1/n) = n√a
a^(m/n) = (n√a)^m = n√(a^m)
   Example: 25^(1/2) = √25 = 5
   Example: 8^(2/3) = (3√8)^2 = (2)^2 = 4

graph TD
    A["Start with an expression"] --> B{Are bases the same?};
    B -->|Yes| C{What operation?};
    C -->|Multiply| D["Add powers: a^(m+n)"];
    C -->|Divide| E["Subtract powers: a^(m-n)"];
    C -->|Power of a power| F["Multiply powers: a^(m*n)"];
    B -->|No (different bases)| G{Is there a 0 or negative power?};
    G -->|Yes (0 power)| H["Result is 1"];
    G -->|Yes (Negative power)| I["Flip to 1/a^n"];
    G -->|No (Fractional power)| J["Use roots: a^(m/n) = (n√a)^m"];
    J --> K["Simplify numerical value"];
    I --> K;
    H --> K;
    F --> K;
    E --> K;
    D --> K;
    K --> L["Final simplified form"];

What are Surds?

A surd is a square root that can't be simplified to a whole number. For example, √9 is 3, so it's not a surd. But √2, √3, √7, √10 are surds because you can't simplify them into a whole number or a clean fraction.

Key Rules for Surds

1. Multiplication: √a * √b = √(a*b)
   Example: √3 * √5 = √15

2. Division: √a / √b = √(a/b)
   Example: √10 / √2 = √(10/2) = √5

3. Simplifying Surds: Look for square factors inside the surd.
   √(a*b) = √a * √b (where a is a perfect square)
   Example: √12 = √(4 * 3) = √4 * √3 = 2√3

4. Adding/Subtracting Surds: You can only add or subtract surds if they have the exact same surd part. Think of √3 as 'x', so 2√3 + 5√3 = 7√3.
   Example: 3√2 + 4√2 = 7√2
   Example: √5 + √2 cannot be simplified further.
   Example: √18 + √8 = √(9*2) + √(4*2) = 3√2 + 2√2 = 5√2

5. Rationalising the Denominator: This means getting rid of the surd in the denominator of a fraction. You do this by multiplying the top and bottom of the fraction by the surd in the denominator.
   Example: 1 / √3 = (1 * √3) / (√3 * √3) = √3 / 3
   Example: 6 / √2 = (6 * √2) / (√2 * √2) = 6√2 / 2 = 3√2

3. Worked Example

Let's simplify the following expression involving both indices and surds:
(4x^3 y^(-2))^2 * √72 / (2x^5 y^(-3))

1. Deal with the first bracket (indices power of a power and negative power):
   (4x^3 y^(-2))^2 = 4^2 * (x^3)^2 * (y^(-2))^2
= 16 * x^(3*2) * y^(-2*2)
= 16x^6 y^(-4)

2. Simplify the surd:
   √72 = √(36 * 2) = √36 * √2 = 6√2

3. Put it all back into the original expression:
   (16x^6 y^(-4) * 6√2) / (2x^5 y^(-3))

4. Group numbers, x terms, y terms, and surd terms:
   (16 * 6 / 2) * (x^6 / x^5) * (y^(-4) / y^(-3)) * √2

5. Simplify each group:

   • Numbers: (16 * 6) / 2 = 96 / 2 = 48
   • x terms (division rule): x^6 / x^5 = x^(6-5) = x^1 = x
   • y terms (division rule): y^(-4) / y^(-3) = y^(-4 - (-3)) = y^(-4 + 3) = y^(-1)
   • Surd: √2

6. Combine everything, making sure to handle the negative power:
   48 * x * y^(-1) * √2
= (48x√2) / y

So the simplified expression is (48x√2) / y.

4. Key Takeaways

• Always apply index rules first before simplifying numbers.
• For multiplication/division with indices, bases must be the same.
• A negative index means the reciprocal; don't confuse it with a negative number.
• A fractional index means a root (e.g., ^(1/2) is square root, ^(1/3) is cube root).
• Simplify surds by finding the largest square factor inside the root.
• You can only add or subtract surds if they have the identical surd part (e.g., √3 and √3).
• Rationalising the denominator removes surds from the bottom of a fraction.

Common Mistakes to Avoid

• Adding powers when multiplying different bases: 2^3 * 3^2 is not 6^(3+2).
• Mixing up (a^m)^n with a^m * a^n: one multiplies powers, the other adds.
• a^0 = 0: Remember, a^0 is always 1 (for a ≠ 0).
• Simplifying √a + √b to √(a+b): This is incorrect; √9 + √16 = 3 + 4 = 7, but √(9+16) = √25 = 5.
• Forgetting to simplify surds before adding/subtracting: √12 + √3 can be simplified to 2√3 + √3 = 3√3.

5. Now Try It

Spend 15 minutes simplifying these two expressions fully.
1. (3a^2 b^(-3))^3 * (a^5 b^4) / (9a^4 b^(-1))
2. √45 + 2√20 - √80

Once you've tried them, check your work. For the first, you should end up with 3a^7 / b^4. For the second, you should get 3√5. Don't worry if it takes a couple of attempts! Focus on applying one rule at a time carefully.