Ordering, standard form, estimation, bounds, indices, surds

Ordering Numbers

Think of numbers as a line of people waiting for a bus. The smallest number is at the front, the largest at the back.

To compare two numbers:

1. Align the decimal points.
2. Start from the leftmost digit and move right.
3. The first different digit tells you which number is larger.

Example: $-5$ vs $-3$ → $-5 < -3$ because -5 is further left on the number line.

Standard Form (Scientific Notation)

Standard form is like a phone number: 1.23 × 105 is easier to read than 123000.

Rules:

• The coefficient is between 1 and 10.
• The exponent is an integer.
• Positive exponent → move decimal right.
• Negative exponent → move decimal left.

Example: $0.00045$ → $4.5 × 10^{-4}$.

Estimation & Bounds

Estimation is like guessing the distance to the next town. You can use bounds to narrow down the answer.

Method:

1. Round each number to a convenient value.
2. Perform the operation with the rounded numbers.
3. Adjust the result to give a range (lower and upper bounds).

Example: Estimate $3.6 × 4.2$.

• Round to $4 × 4 = 16$ (lower bound).
• Round to $5 × 5 = 25$ (upper bound).
• So $16 < 3.6 × 4.2 < 25$.

Indices (Powers & Roots)

Indices are like a recipe: 3² means “three times three.”

Key rules:

• $a^m × a^n = a^{m+n}$
• $(a^m)^n = a^{mn}$
• $a^0 = 1$ (for $a ≠ 0$)
• $a^{-n} = \frac{1}{a^n}$

Example: $2^3 × 2^4 = 2^{3+4} = 2^7 = 128$.

Surds (Radicals)

Surds are numbers that cannot be simplified to a whole number, like the square root of 2.

Rules for simplifying:

• Factor inside the radical into perfect squares.
• Take the square root of the perfect square out of the radical.

Example: $\sqrt{72}$.

1. Factor: $72 = 36 × 2$.
2. Take out: $\sqrt{36} × \sqrt{2} = 6\sqrt{2}$.

When multiplying surds: $\sqrt{a} × \sqrt{b} = \sqrt{ab}$.

Exam Success Checklist

• ?? Check your work: Verify calculations with bounds or estimation.
• ?? Use standard form: Convert large numbers to keep calculations manageable.
• ?? Simplify surds: Always reduce to simplest form.
• ?? Show all steps: Even if you know the answer, write the process.
• ?? Time management: Allocate time for checking and re‑calculating.

Good luck, and remember: practice makes perfect! 🚀