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 are bounds , in the context of estimation?

Bounds are the smallest ( lower bound ) and largest ( upper bound ) numbers that a rounded number can lie between .

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.

Define the term lower bound .

The lower bound is the smallest possible value a number could have been before rounding .

True or False? A rounded number could have been equal to its upper bound before rounding.

False. A rounded number can not be equal to its upper bound before rounding. The upper bound of a rounded number is the smallest value that would have rounded up to the next highest rounded value.

True or False? The basic rule for finding bounds is " half up, half down ".

True. The basic rule for finding bounds is " half up, half down ". E.g. if a number is rounded to 31 to the nearest integer , then halfway down to the next lowest integer is 30.5 ( lower bound ), halfway up to the next highest integer is 31.5 ( upper bound ).

True or False? Percentage error should always be a positive number.

True. Percentage error should always be a positive number.

Define the term significant figures .

Significant figures are the digits in a number that carry meaning contributing to its precision. Significant figures start with the first non-zero digit .

True or False? When rounding to 3 significant figures , you always keep the first three non-zero digits .

False. When rounding to 3 significant figures , you keep the first three significant digits , which may include zeroes after the first non-zero digit.

What is the purpose of estimation ?

The purpose of estimation is to find approximate answers to difficult sums or to check if answers are about the right size (order of magnitude).

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 .

True or False? When dealing with currency , you should always round to the nearest whole number .

False. The appropriate rounding for currency depends on the specific currency being used. Some are rounded to whole numbers , others to 2 decimal places .

What is a logarithm ?

A logarithm is the inverse of an exponent .

What is the rule for rounding up in problems involving minimum number of objects ?

In problems involving minimum number of objects , always round up to ensure there are enough objects to meet the requirement. E.g. if you calculate that 13.23 tins of paint are needed to complete a job, round up to 14 tins to make sure enough paint is bought.

Define the term system of linear equations .

A system of linear equations is where two or more linear equations work together , with variables that satisfy all equations simultaneously.

True or False? A system of linear equations may also be referred to as simultaneous equations .

True. A system of linear equations may also be referred to as simultaneous equations .

True or False? You can use a GDC to solve a system of linear equations .

True. You can use a GDC to solve a system of linear equations . Your GDC will have a simultaneous equations solving feature. However it is also useful to be able to solve simple simultaneous equations 'by hand'.

True or False? A polynomial equation of order five can have up to five solutions .

True. A polynomial equation of order five can have up to five solutions . In general, a polynomial equation of order n can have up to n (unique) solutions .

True or False? A polynomial equation of an odd degree will always have at least one solution .

True. A polynomial equation of an odd degree will always have at least one solution . (It is possible that a polynomial equation of an even degree will have no real-number solutions .)

List two ways that you can use a GDC to solve polynomial equations .

To use a GDC to solve polynomial equations you can: Use the equation solving feature(s) of your GDC to solve the equation directly. Use the graphing mode to visualise the equation, then use the analyse function to find the solutions (zeros or roots).

Define the term root in the context of polynomial equations .

In the context of polynomial equations , a root is a value of the variable that makes the polynomial equal to zero , also called a solution or zero of the equation.

True or False? A GDC will always show all solutions to a polynomial equation.

False. Some GDCs may only show the first solution to a polynomial equation, so it can important to graph the function to check how many solutions there should be.

What is the relationship between the degree of a polynomial and its graph ?

The degree of a polynomial determines the maximum number of times its graph can cross the x -axis . This corresponds to the maximum number of real roots .

Define the term zero in the context of polynomial equations .

In the context of polynomial equations , a zero is a value of the variable that makes the polynomial equal to zero , also called a root or solution .

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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 are bounds, in the context of estimation?
Bounds are the smallest (lower bound) and largest (upper bound) numbers that a rounded number can lie between.
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.
Define the term lower bound.
The lower bound is the smallest possible value a number could have been before rounding.
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. $6^{1} = 6$ .
True or False?
A rounded number could have been equal to its upper bound before rounding.
False.
A rounded number can not be equal to its upper bound before rounding.
The upper bound of a rounded number is the smallest value that would have rounded up to the next highest rounded value.
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. $3^{- 1} = \frac{1}{3}$ .
True or False?
The basic rule for finding bounds is "half up, half down".
True.
The basic rule for finding bounds is "half up, half down".
E.g. if a number is rounded to 31 to the nearest integer, then
- halfway down to the next lowest integer is 30.5 (lower bound),
- halfway up to the next highest integer is 31.5 (upper bound).
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 formula for percentage error?
The formula for percentage error is $ε = |\frac{v_{A} - v_{E}}{v_{E}}| \times 100 %$
Where:
- $ν_{A}$ is the approximate value
- $ν_{E}$ is the exact value
This formula is contained in the exam formula booklet.
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.
True or False?
Percentage error should always be a positive number.
True.
Percentage error should always be a positive number.
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 are exact values?
Exact values are forms that represent the full and precise value of a number, often involving fractions, roots (or surds), logarithms, or mathematical constants (e.g. $π$ ).
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.
Define the term significant figures.
Significant figures are the digits in a number that carry meaning contributing to its precision.
Significant figures start with the first non-zero digit.
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.
True or False?
When rounding to 3 significant figures, you always keep the first three non-zero digits.
False.
When rounding to 3 significant figures, you keep the first three significant digits, which may include zeroes after the first non-zero digit.
Define the term reciprocal in relation to exponents.
In relation to exponents, the reciprocal of $x^{m}$ is $x^{- m}$ , which equals $\frac{1}{x^{m}}$ .
What is the purpose of estimation?
The purpose of estimation is to find approximate answers to difficult sums or to check if answers are about the right size (order of magnitude).
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.
True or False?
When dealing with currency, you should always round to the nearest whole number.
False.
The appropriate rounding for currency depends on the specific currency being used. Some are rounded to whole numbers, others to 2 decimal places.
What is a logarithm?
A logarithm is the inverse of an exponent.
What is the rule for rounding up in problems involving minimum number of objects?
In problems involving minimum number of objects, always round up to ensure there are enough objects to meet the requirement.
E.g. if you calculate that 13.23 tins of paint are needed to complete a job, round up to 14 tins to make sure enough paint is bought.
State the logarithm equation in terms of $a$ , $b$ , and $x$ that is equivalent to $a^{x} = b$ .
If $a^{x} = b$ , then the equivalent logarithm equation is $\log_{a} b = x$ .
This is valid so long as $a > 0$ , $b > 0$ and $a \neq 1$ .
This logarithm equation is given in the exam formula booklet.
Define the term absolute value.
Absolute value is the non-negative value of a number without regard to its sign, denoted by vertical bars | | around the number.
Absolute value is sometimes also referred to as the modulus of a number.
E.g. the absolute value of 3 is 3, i.e. $|3| = 3$ .
The absolute value of -7 is 7, i.e. $|- 7| = 7$ .
What is $\ln x$ the notation for?
$\ln x$ is the notation for the natural logarithm of $x$ .
This is equivalent to $\log_{e} x$ , where $e$ is the mathematical constant approximately equal to 2.718.
True or False?
$\log x$ is sometimes used as an abbreviation for $\log_{10} x$ .
True.
$\log x$ is sometimes used as an abbreviation for $\log_{10} x$ .
$\log x$ will usually mean $\log_{10} x$ , unless otherwise specified.
Define the term base in the context of logarithms.
In the context of logarithms, the base is the number that is being raised to a power in the equivalent exponential equation.
E.g. in $\log_{a} b$ the base is $a$ .
True or False?
The equation $2^{x} = 10$ can be solved using logarithms.
True.
The equation $2^{x} = 10$ can be solved using logarithms, specifically by finding the value of the solution $x = \log_{2} 10$ .
What is a linear equation?
A linear equation is an equation of the first order (degree 1), usually written in the form $a x + b y + c = 0$ where $a$ , $b$ , and $c$ are constants.
Define the term system of linear equations.
A system of linear equations is where two or more linear equations work together, with variables that satisfy all equations simultaneously.
True or False?
A system of linear equations may also be referred to as simultaneous equations.
True.
A system of linear equations may also be referred to as simultaneous equations.
True or False?
You can use a GDC to solve a system of linear equations.
True.
You can use a GDC to solve a system of linear equations.
Your GDC will have a simultaneous equations solving feature. However it is also useful to be able to solve simple simultaneous equations 'by hand'.
What is a polynomial?
A polynomial is an algebraic expression consisting of a finite number of terms, with non-negative integer indices only.
E.g. $3 x^{2} - 5 x + 17$ is a polynomial.
What is a polynomial equation?
A polynomial equation is an equation where a polynomial is set equal to zero.
E.g. $3 x^{2} - 5 x + 17 = 0$ is a polynomial equation.
True or False?
A polynomial equation of order five can have up to five solutions.
True.
A polynomial equation of order five can have up to five solutions.
In general, a polynomial equation of order n can have up to n (unique) solutions.
True or False?
A polynomial equation of an odd degree will always have at least one solution.
True.
A polynomial equation of an odd degree will always have at least one solution.
(It is possible that a polynomial equation of an even degree will have no real-number solutions.)
List two ways that you can use a GDC to solve polynomial equations.
To use a GDC to solve polynomial equations you can:
1. Use the equation solving feature(s) of your GDC to solve the equation directly.
2. Use the graphing mode to visualise the equation, then use the analyse function to find the solutions (zeros or roots).
Define the term root in the context of polynomial equations.
In the context of polynomial equations, a root is a value of the variable that makes the polynomial equal to zero, also called a solution or zero of the equation.
True or False?
A GDC will always show all solutions to a polynomial equation.
False.
Some GDCs may only show the first solution to a polynomial equation, so it can important to graph the function to check how many solutions there should be.
What is the relationship between the degree of a polynomial and its graph?
The degree of a polynomial determines the maximum number of times its graph can cross the x-axis.
This corresponds to the maximum number of real roots.
Define the term zero in the context of polynomial equations.
In the context of polynomial equations, a zero is a value of the variable that makes the polynomial equal to zero, also called a root or solution.