## Podcast Beta

## Questions and Answers

What is the prime factor decomposition of 56?

What is the area of a trapezium with bases of lengths 8 and 5 and a height of 4?

Which of the following is equivalent to the expression 3(x + 4) + 2(x - 2)?

What is the surface area of a cube with a side length of 3 units?

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Which equation correctly shows the solution to 2x + 5 = 15?

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## Study Notes

### Prime Factor Decomposition

- Finding the prime numbers that multiply together to make a given number
- A prime number can only be divided by 1 and itself.
- Example: 24 = 2 x 2 x 2 x 3

### Laws of Indices

- Explain how to multiply and divide numbers with powers
- Example: x^m x x^n = x^(m+n)
- Example: x^m / x^n = x^(m-n)

### Powers of 10

- Used to express very large or very small numbers in a more concise way
- Uses powers of 10 to represent the number of digits in a number
- Example: 1000 = 10^3, 0.01 = 10^-2

### Rounding and Estimating

- Approximating a number to the nearest whole number or decimal place
- Used for making calculations easier
- Example: 12.34 rounded to the nearest whole number is 12

### Simplifying Expressions

- Combining like terms and using laws of indices
- Goal: to make the expression easier to understand
- Example: 2x + 3x - 5 = 5x - 5

### Expanding and Factorising

- Expanding brackets: Multiplying each term inside the brackets by the term outside the brackets
- Factorising: Writing an expression as a product of two or more factors
- Example: Expanding (x + 2)(x - 1) = x^2 + x - 2
- Example: Factorising x^2 + 5x + 6 = (x + 2)(x + 3)

### Substituting

- Replacing variables in an expression with given values
- Evaluates the expression to find the answer
- Example: If x = 2, then 3x + 1 = 3(2) + 1 = 7

### Solving

- Finding the value of a variable that makes an equation true.
- Isolate the variable on one side of the equation by performing the same operation on both sides
- Example: To solve for x in 2x + 3 = 7, we can subtract 3 from both sides to get 2x = 4, then divide both sides by 2 to get x = 2

### Solving One-Step Equations

- Equations that require just one arithmetic step
- Example: x + 5 = 8; Subtract 5 from both sides to get x = 3

### Solving Two-Step Equations

- Equations that require two arithmetic steps in a row
- Example: 2x + 5 = 11. Subtract 5 from both sides to get 2x = 6, then divide both sides by 2 to get x = 3

### More Complex Equations

- Equations involving more than one variable
- Involve rearranging to solve for specific variables.
- Example: To solve for y in 2x + 3y = 12, subtract 2x from both sides to get 3y = 12 - 2x, then divide both sides by 3 to get y = (12 - 2x) / 3

### Angles in Polygons

- The sum of the interior angles of a polygon can be found using the formula (n-2) x 180 degrees, where 'n' is the number of sides
- Example: The sum of the interior angles of a hexagon (6 sides) is (6-2) x 180 = 720 degrees

### Triangles, Parallelograms and Trapeziums

- Triangles: Have three sides and three angles
- Parallelograms: Have four sides with two sets of parallel sides
- Trapeziums: Have four sides where only one pair of sides is parallel
- Example: The area of a triangle is (1/2) x base x height

### Perimeter and Area of Compound Shapes

- Perimeter is the total length of the outside edges of a shape
- Area is the amount of space inside a shape
- Compound shapes can be divided into simpler shapes to calculate their area and perimeter
- Example: A compound shape made up of a rectangle and a semicircle can have its area calculated by adding the area of the rectangle and the area of the semicircle

### Surface Area

- The total area of all the faces of a 3-dimensional object
- Example: The surface area of a cube with side length 's' is 6s^2

### Volume

- The amount of space a 3-dimensional object takes up.
- Example: The volume of a rectangular prism is length x width x height

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## Description

Test your understanding of key algebraic concepts such as prime factor decomposition, laws of indices, and simplifying expressions. This quiz covers essential techniques for manipulating numbers and expressions, including rounding and estimating. Prepare to enhance your algebra skills with various examples and practical applications.