Year 9 Higher Math — 8-topic bundle

Example:
3(x + 5)

Here, you multiply 3 by x and 3 by 5.
3 * x = 3x
3 * 5 = 15
So, 3(x + 5) = 3x + 15

Example:
(x + 2)(x + 3)

1. First: Multiply the first terms in each bracket: x * x = x²
2. Outer: Multiply the outer terms: x * 3 = 3x
3. Inner: Multiply the inner terms: 2 * x = 2x
4. Last: Multiply the last terms in each bracket: 2 * 3 = 6

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

You already know the four basic operations: addition, subtraction, multiplication, and division. We'll quickly review some key ideas about them and then dive into factors and primes.

• Brackets first
• Indices (powers/roots)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

For example, $2 + 3 \times 4$ is not $(2+3) \times 4 = 20$. It's $2 + (3 \times 4) = 2 + 12 = 14$.

When a number is rounded, it means we don't know its exact value, only that it falls within a certain range. We use bounds to define this range.

Let's say a number has been rounded to a certain degree of accuracy (e.g., to the nearest whole number, to one decimal place, or to the nearest 10).

• Lower Bound (LB): This is the smallest possible value the original number could have been before rounding up to the given rounded value.
• Upper Bound (UB): This is the largest possible value the original number could have been before rounding down to the given rounded value.

Let's look at the basic operations and their inverses:
* Addition (+) and Subtraction (-) are inverses.
* Multiplication (×) and Division (÷) are inverses.
* Squaring (x²) and Square Root (√) are inverses.

When you want to solve for x, you need to "undo" all the operations happening to x in the reverse order they were applied, using their inverse operations.

1. Addition/Subtraction: Deal with any terms being added or subtracted away from the variable you want to isolate.
2. Multiplication/Division: Next, handle any terms multiplying or dividing the variable.
3. Indices/Roots: Finally, take care of any powers or roots applied directly to the variable.
4. Brackets: If the variable is inside a bracket, you often need to expand the bracket first, or treat the entire bracket as a single term until other operations are undone.

Here are the main index laws you need to know:

1. Multiplication Law: When multiplying numbers with the same base, you add the powers.
   $a^m \times a^n = a^{m+n}$
   Example: $2^3 \times 2^4 = 2^{3+4} = 2^7 = 128$

2. Division Law: When dividing numbers with the same base, you subtract the powers.
   $a^m \div a^n = a^{m-n}$
   Example: $5^6 \div 5^2 = 5^{6-2} = 5^4 = 625$

Let's start with the simplest case: quadratics where $a=1$, like $x^2 + bx + c$.