Foundation plus - Numerical operations - GCSE Mathematics

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Foundation plus - Numerical operations

infoWhy this? Efficient written methods and accurate calculator use help us carry out and check complex calculations. Indices, standard form, and surds extend the numbers we can handle, allowing us to represent extreme magnitudes, work with roots, and distinguish exact values from approximations.

scheduleWhy now? This unit revisits the powers and factors studied in autumn and extends the index laws already used in algebraic manipulation. It prepares us to use powers in sequence rules and retain exact values in later quadratic and trigonometric calculations.

The formal written method for multiplication (long multiplication) can be used to multiply numbers up to 4 digits by a two-digit number.
The formal written method for division (long division) can be used to divide numbers up to 4 digits by a two-digit number.
Calculators follow the order of operations (BIDMAS) when performing calculations.
The laws of indices include:
( am x an = am+n )
( am ÷ an = am-n )
( (a(m)^n) = am x n )
( a0 = 1 ) (for ( a ≠ 0 ))
(a1 = a )
The laws of indices apply to algebraic bases as well as numerical bases.
Standard form is a way of writing very large or very small numbers as ( a x 10n ), where ( 1 ≤ a < 10) and (n) is an integer.
Large numbers in standard form have positive powers of 10; small numbers (less than 1) have negative powers of 10.
To convert a number from standard form to an ordinary number, multiply the coefficient by the appropriate power of 10.
To order numbers in standard form, compare the powers of 10 first, then the coefficients if the powers are equal.
When multiplying numbers in standard form, multiply the coefficients and add the powers of 10.
When dividing numbers in standard form, divide the coefficients and subtract the powers of 10.
When adding or subtracting numbers in standard form, the powers of 10 must be the same.
Fractional indices represent roots, e.g., ( a^{1/n} = √(n&a)) and ( a^{m/n} = √(n&(a)^m))
Negative indices represent reciprocals, e.g., a-n) = 1/an ).
Surds are irrational roots that cannot be simplified to a rational number (e.g., (√2), and can be simplified by factoring out square numbers.
Estimating powers and roots involves finding two consecutive integers between which the answer lies

rocket_launchYou must be able to

Multiply numbers up to 4 digits by a two-digit number using the formal written (long multiplication) method.
Divide numbers up to 4 digits by a two-digit number using the formal written (long division) method.
Use a calculator accurately to perform multi-step calculations, ensuring correct use of brackets and order of operations.
Apply the laws of indices to simplify expressions with numerical and algebraic bases.
Write large or small numbers in standard form and convert numbers in standard form to ordinary numbers.
Order a list of numbers given in standard form.
Multiply and divide numbers in standard form, expressing the answer in standard form.
Add and subtract numbers in standard form, expressing the answer in standard form.
Calculate with fractional indices, including evaluating expressions such as ( 16^{3/4} ).
Estimate the value of powers and roots of positive numbers (e.g., estimate ( 5^{1.7} ) or ( \sqrt[3]{50} )).
Evaluate expressions with negative indices.
Simplify surds, including rationalising denominators and expressing surds in their simplest form.

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