3.1 Basic Manipulation
Rules for Negative Numbers
Negative numbers can make you go from being correct to very wrong, you have to keep track of all negatives.
Negative numbers can make you go from being correct to very wrong, you have to keep track of all negatives.
⚡ Multiplication & Division Rules:
The same rules apply for division!
| Operation | Result | Example |
|---|---|---|
| $(+) \times (+)$ | $+$ | $3 \times 4 = 12$ |
| $(+) \times (-)$ | $-$ | $3 \times (-4) = -12$ |
| $(-) \times (+)$ | $-$ | $(-3) \times 4 = -12$ |
| $(-) \times (-)$ | $+$ | $(-3) \times (-4) = 12$ |
The same rules apply for division!
💡 How to remember
• Same signs → Positive result
• Different signs → Negative result
• Different signs → Negative result
Addition & Subtraction with Negatives:
Subtracting a negative = Adding:
$$5 - (-3) = 5 + 3 = 8$$
Adding a negative = Subtracting:
$$5 + (-3) = 5 - 3 = 2$$
Double negative:
$$-(-7) = +7$$
$$5 - (-3) = 5 + 3 = 8$$
Adding a negative = Subtracting:
$$5 + (-3) = 5 - 3 = 2$$
Double negative:
$$-(-7) = +7$$
⚠️ Watch Out: Negatives and Powers
$(-5)^2 = (-5) \times (-5) = 25$ ✓ brackets mean the negative is squared
$-5^2 = -(5 \times 5) = -25$ ✗ only the 5 is squared, then made negative
$-(5)^2 = -(25) = -25$ ✗ same as above
$-5^2 = -(5 \times 5) = -25$ ✗ only the 5 is squared, then made negative
$-(5)^2 = -(25) = -25$ ✗ same as above
🧮 Negative Number Calculator
🎯 Negative Number Practice
Real Life Uses:
• Temperature: Below zero readings and temperature changes
• Finance: Debts, losses, and negative balances
• Elevation: Below sea level measurements
• Science: Negative charges, vectors, and directions
• Temperature: Below zero readings and temperature changes
• Finance: Debts, losses, and negative balances
• Elevation: Below sea level measurements
• Science: Negative charges, vectors, and directions
What is Algebra?
In algebra, we use letters to represent numbers. This allows us to write general rules and solve problems where we don't know all the values yet.
A term is a group of numbers and letters multiplied or divided together. Terms are separated by $+$ or $-$ signs.
Like terms have exactly the same letters with the same powers.
Unlike terms have different letters or different powers.
To simplify an expression, add or subtract the coefficients (numbers in front) of like terms.
In algebra, we use letters to represent numbers. This allows us to write general rules and solve problems where we don't know all the values yet.
⚡ Algebraic Notation:
Terms in Algebra
• $xyz = x \times y \times z$ (letters next to each other means multiply)
• $3x = 3 \times x$ (number in front of letter means multiply)
• $xy^2 = x \times y \times y$ (only the $y$ is squared)
• $(xy)^2 = x \times x \times y \times y = x^2y^2$ (both are squared)
• $x(y+z)^3 = x \times (y+z) \times (y+z) \times (y+z)$
• $3x = 3 \times x$ (number in front of letter means multiply)
• $xy^2 = x \times y \times y$ (only the $y$ is squared)
• $(xy)^2 = x \times x \times y \times y = x^2y^2$ (both are squared)
• $x(y+z)^3 = x \times (y+z) \times (y+z) \times (y+z)$
A term is a group of numbers and letters multiplied or divided together. Terms are separated by $+$ or $-$ signs.
Example: Identify the terms in $3x + 4y - 5z$
Like Terms and Unlike Terms
This expression has three terms:
• $3x$ (first term)
• $4y$ (second term)
• $-5z$ (third term — the negative sign belongs to it!)
• $3x$ (first term)
• $4y$ (second term)
• $-5z$ (third term — the negative sign belongs to it!)
Like terms have exactly the same letters with the same powers.
Unlike terms have different letters or different powers.
✅ Like Terms (can be combined):
• $3x$ and $5x$ → both have just $x$
• $2xy$ and $4yx$ → both have $xy$ (order doesn't matter)
• $5x^2$ and $-3x^2$ → both have $x^2$
• $7ab^2$ and $2ab^2$ → both have $ab^2$
• $2xy$ and $4yx$ → both have $xy$ (order doesn't matter)
• $5x^2$ and $-3x^2$ → both have $x^2$
• $7ab^2$ and $2ab^2$ → both have $ab^2$
❌ Unlike Terms (cannot be combined):
Collecting Like Terms
• $3x$ and $4y$ → different letters
• $2x^2$ and $3x$ → different powers of $x$
• $5xy$ and $5x^2y$ → different: $xy$ vs $x^2y$
• $4a$ and $4a^2$ → different powers of $a$
• $2x^2$ and $3x$ → different powers of $x$
• $5xy$ and $5x^2y$ → different: $xy$ vs $x^2y$
• $4a$ and $4a^2$ → different powers of $a$
To simplify an expression, add or subtract the coefficients (numbers in front) of like terms.
Examples:
$$3x + 5x = 8x$$
$$7y - 2y = 5y$$
$$4x^2 + 3x^2 - x^2 = 6x^2$$
$$5a + 3b - 2a + b = 3a + 4b$$
Worked Example: Simplify $4x + 3y - 2x + 5y - x$
Step 1: Group like terms
$= (4x - 2x - x) + (3y + 5y)$
Step 2: Combine like terms
$= x + 8y$
$= (4x - 2x - x) + (3y + 5y)$
Step 2: Combine like terms
$= x + 8y$
🎯 Practice: Like or Unlike Terms?
🎯 Practice: Simplify by Collecting Like Terms
Real Life Uses:
• Formulas: Area = $l \times w$, Distance = Speed × Time
• Programming: Variables in code work just like algebra
• Finance: Calculating totals with unknown quantities
• Science: Physics equations like $F = ma$
• Formulas: Area = $l \times w$, Distance = Speed × Time
• Programming: Variables in code work just like algebra
• Finance: Calculating totals with unknown quantities
• Science: Physics equations like $F = ma$
What are Indices?
Indices (singular: index) are the small numbers written above and to the right of a base number, also called powers or exponents.
Fractional indices represent roots.
Indices (singular: index) are the small numbers written above and to the right of a base number, also called powers or exponents.
Basic Notation:
$$a^n = \underbrace{a \times a \times a \times ... \times a}_{n \text{ times}}$$
• $a$ is the base
• $n$ is the index/power/exponent
• $a$ is the base
• $n$ is the index/power/exponent
⚡ The Laws of Indices:
Fractional Indices| Law | Rule | Example |
|---|---|---|
| Multiplication | $a^m \times a^n = a^{m+n}$ | $x^3 \times x^4 = x^7$ |
| Division | $a^m \div a^n = a^{m-n}$ | $x^5 \div x^2 = x^3$ |
| Power of a Power | $(a^m)^n = a^{m \times n}$ | $(x^2)^3 = x^6$ |
| Power of Zero | $a^0 = 1$ | $5^0 = 1$ |
| Power of One | $a^1 = a$ | $7^1 = 7$ |
| Negative Index | $a^{-n} = \frac{1}{a^n}$ | $x^{-2} = \frac{1}{x^2}$ |
| Power of Product | $(ab)^n = a^n \times b^n$ | $(2x)^3 = 8x^3$ |
| Power of Quotient | $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ | $\left(\frac{x}{2}\right)^2 = \frac{x^2}{4}$ |
Fractional indices represent roots.
💡 Fractional Index Rule:
$$a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$$
• The denominator is the root
• The numerator is the power
• The denominator is the root
• The numerator is the power
Examples of Fractional Indices:
$$x^{\frac{1}{2}} = \sqrt{x}$$
$$x^{\frac{1}{3}} = \sqrt[3]{x}$$
$$8^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$$
$$27^{\frac{2}{3}} = \left(\sqrt[3]{27}\right)^2 = 3^2 = 9$$
$$16^{\frac{3}{4}} = \left(\sqrt[4]{16}\right)^3 = 2^3 = 8$$
Worked Example: Simplify $\frac{x^5 \times x^3}{x^2}$
Step 1: Use multiplication law on numerator
$$= \frac{x^{5+3}}{x^2} = \frac{x^8}{x^2}$$
Step 2: Use division law
$$= x^{8-2} = x^6$$
$$= \frac{x^{5+3}}{x^2} = \frac{x^8}{x^2}$$
Step 2: Use division law
$$= x^{8-2} = x^6$$
Worked Example: Simplify $(2x^3)^4$
Step 1: Apply power to each part
$$= 2^4 \times (x^3)^4$$
Step 2: Calculate
$$= 16 \times x^{12} = 16x^{12}$$
$$= 2^4 \times (x^3)^4$$
Step 2: Calculate
$$= 16 \times x^{12} = 16x^{12}$$
🎯 Index Law Practice
🎯 Evaluate Fractional Indices
Real Life Uses:
• Science: Scientific notation uses powers of 10
• Computing: Binary (powers of 2), data storage
• Finance: Compound interest formulas
• Physics: Inverse square laws, exponential decay
• Biology: Population growth, bacterial doubling
• Science: Scientific notation uses powers of 10
• Computing: Binary (powers of 2), data storage
• Finance: Compound interest formulas
• Physics: Inverse square laws, exponential decay
• Biology: Population growth, bacterial doubling