What is the advantage of using standard form for very large or small numbers?

Standard form lets us represent very large or very small numbers in a concise and manageable way using powers of 10. This means we can write them more neatly, compare them more easily, and carry out calculations more efficiently.

True or False? There is always one non – zero digit before the decimal point in a standard form number.

True. There is always one (and only one ) non – zero digit before the decimal point in a standard form number.

True or False? Scientific notation is another term for standard form.

True. Scientific notation is another term for standard form.

True or False? The exponent k in standard form must always be positive .

False. The exponent k in standard form can be positive , negative , or zero .

Define the term exponent .

An exponent is a power that a number (called the base ) is raised to.

What number do you get when you raise any non-zero number to the power of zero , e.g. 2 0 ?

Any non-zero number raised to the power of 0 is equal to 1 . E.g. 2 0 = 1.

True or False? Index laws only work with terms that have the same base .

True. Index laws only work with terms that have the same base .

What is the first step in finding partial fractions if the denominator is a quadratic ?

The first step in finding partial fractions if the denominator is a quadratic is to factorise the denominator . It is a good idea at this point, to also check to see if there are any common factors that can be cancelled.

True or False? Partial fractions can only be used with rational functions that have linear or quadratic denominators .

False. Partial fractions can be used with rational functions that have linear or quadratic denominators but also for higher degree polynomials that can be factorised .

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What is the advantage of using standard form for very large or small numbers?

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What is the advantage of using standard form for very large or small numbers?
Standard form lets us represent very large or very small numbers in a concise and manageable way using powers of 10. This means we can write them more neatly, compare them more easily, and carry out calculations more efficiently.
True or False?
In standard form, numbers are always written in the form $a \times 10^{k}$ , where $1 \leq a \leq 10$ and $k$ is an integer.
False.
In standard form, numbers are always written in the form $a \times 10^{k}$ , where $1 \leq a < 10$ (not $1 \leq a \leq 10$ ) and $k$ is an integer.
In standard form, the value of a must be greater than or equal to 1 and less than 10 .
True or False?
There is always one non – zero digit before the decimal point in a standard form number.
True.
There is always one (and only one) non – zero digit before the decimal point in a standard form number.
What does $a E n$ mean on a calculator display?
On a calculator display, $a E n$ means $a \times 10^{n}$ in standard form.
(Some calculators use that form of notation instead of the usual standard form notation.)
True or False?
Scientific notation is another term for standard form.
True.
Scientific notation is another term for standard form.
True or False?
The exponent k in standard form must always be positive.
False.
The exponent k in standard form can be positive, negative, or zero.
Define the term exponent.
An exponent is a power that a number (called the base) is raised to.
What number do you get when you raise any non-zero number to the power of zero, e.g. 2⁰?
Any non-zero number raised to the power of 0 is equal to 1.
E.g. 2⁰ = 1.
True or False?
If you raise a non-zero number to the power of 1, you get 1.
False.
Any number raised to the power 1 is just itself.
E.g. $6^{1} = 6$ .
What do you get if you raise a non-zero number to the power of -1,
e.g. 3^-1?
If you raise a non-zero number to the power of -1 you get the reciprocal of the number.
E.g. $3^{- 1} = \frac{1}{3}$ .
What do you get if you raise a positive number to the power of ½,
e.g. 5^1/2?
If you raise a non-zero number to the power of ½ you get its positive square root.
E.g. $5^{\frac{1}{2}} = \sqrt{5}$ .
What is the index law for $x^{m} \times x^{n}$ ?
$x^{m} \times x^{n} = x^{m + n}$
If you multiply two powers with the same base number, you add the indices together.
This formula is not given in the exam formula booklet.
What is the index law for $x^{m} \div x^{n}$ ?
$x^{m} \div x^{n} = x^{m - n}$
If you divide two powers with the same base number, you subtract one index from the other.
This formula is not given in the exam formula booklet.
What is the index law for ${(x^{m})}^{n}$ ?
${(x^{m})}^{n} = x^{m \times n}$
If you raise a power to another power, you multiply the indices.
This formula is not given in the exam formula booklet.
What is the index law for ${(x y)}^{m}$ ?
${(x y)}^{m} = x^{m} y^{m}$
A power outside brackets is applied to each factor inside the brackets individually.
This formula is not given in the exam formula booklet.
But note that ${(x + y)}^{m} \neq x^{m} + y^{m}$ , i.e. you can only use this index law when the things inside the bracket are multiplied together.
Define the term reciprocal in relation to exponents.
In relation to exponents, the reciprocal of $x^{m}$ is $x^{- m}$ , which equals $\frac{1}{x^{m}}$ .
True or False?
Index laws only work with terms that have the same base.
True.
Index laws only work with terms that have the same base.
If a rational function has a linear numerator and denominator, how can it be rewritten as partial fractions?
If a rational function has a linear numerator and denominator, it can be rewritten as the sum of a constant and a fraction with a linear denominator, e.g. $\frac{a x + b}{c x + d} = A + \frac{B}{c x + d}$
How can a rational function with a quadratic denominator be rewritten using partial fractions?
A rational function with a quadratic denominator can be rewritten as the sum of two rational functions with linear denominators (if the quadratic denominator can be factorised into two distinct factors).
E.g. $\frac{a x + b}{(c x + d) (e x + f)} = \frac{A}{c x + d} + \frac{B}{e x + f}$
What is the first step in finding partial fractions if the denominator is a quadratic?
The first step in finding partial fractions if the denominator is a quadratic is to factorise the denominator.
It is a good idea at this point, to also check to see if there are any common factors that can be cancelled.
What form does a rational expression $\frac{p}{{(a x + b)}^{2}}$ take in partial fractions?
A rational expression $\frac{p}{{(a x + b)}^{2}}$ takes the form $\frac{A}{(a x + b)} + \frac{B}{{(a x + b)}^{2}}$ in partial fractions.
After eliminating fractions, e.g. $\frac{5 x + 5}{(x + 3) (x - 2)} \equiv \frac{A}{x + 3} + \frac{B}{x - 2}$ becomes $5 x + 5 \equiv A (x - 2) + B (x + 3)$
What two methods can be used to find the numerators $A$ and $B$ ?
After eliminating fractions, the two methods that can be used to find the numerators $A$ and $B$ are:
(a) Substitute the root of each linear factor into the identity and solve.
$5 (2) + 5 \equiv A ((2) - 2) + B ((2) + 3)$ and $5 (- 3) + 5 \equiv A ((- 3) - 2) + B ((- 3) + 3)$
(b) Expand brackets and compare coefficients.
$5 x + 5 \equiv (A + B) x + (- 2 A + 3 B)$
True or False?
Partial fractions can only be used with rational functions that have linear or quadratic denominators.
False.
Partial fractions can be used with rational functions that have linear or quadratic denominators but also for higher degree polynomials that can be factorised.

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Number & Algebra Toolkit (DP IB Analysis & Approaches (AA))

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1. Number & Algebra

Number & Algebra Toolkit

Exponentials & Logs

Sequences & Series

Simple Proof & Reasoning

Proof by Induction & Contradiction

Binomial Theorem

Permutations & Combinations

Complex Numbers

Further Complex Numbers

Systems of Linear Equations

2. Functions

Linear Functions & Graphs

Quadratic Functions & Graphs

Functions Toolkit

Other Functions & Graphs

Reciprocal & Rational Functions

Transformations of Graphs

Polynomial Functions

Inequalities

Modulus Functions & Further Transformations

3. Geometry & Trigonometry

Geometry Toolkit

Geometry of 3D Shapes

Trigonometry

The Unit Circle & Exact Values

Trigonometric Functions & Graphs

Trigonometric Equations & Identities

Inverse & Reciprocal Trigonometric Functions

Trigonometric Proof & Equation Strategies

Vector Properties

Vector Equations of Lines

Vector Planes

4. Statistics & Probability

Statistics Toolkit

Correlation & Regression

Probability

Discrete Random Variables

Binomial Distribution

Normal Distribution

Continuous Random Variables

5. Calculus

Differentiation

Techniques & Applications of Differentiation

Integration

Techniques & Applications of Integration

Optimisation

Kinematics

Basic Limits & Continuity

Further Differentiation

Further Integration

Differential Equations

Maclaurin Series

Limits using l'Hôpital's Rule & Maclaurin Series