Laws of indices of all rational exponents[edit | edit source]
An index is of the form , and the laws on how to manipulate them is vital knowledge.
Power of one and zero[edit | edit source]
Any nonzero base raised to the power of one is simply the base.
Any nonzero base raised to the power of zero is one.
has no meaning.
Multiplication and division[edit | edit source]
When two indices are multiplied, as long as the bases are equal, the two indices are simply added together.
For division the opposite is true, as long as the bases are equal, the indices are subtracted.
Fractional indices[edit | edit source]
The denominator of a fractional index is the root that the base must be taken to.
The numerator of a fractional index is the power the base must be raised to.
Negative indices[edit | edit source]
A negative sign for an index shows that the base is the denominator of a fraction, the index is the power it must be raised to. The numerator of this fraction is one (though if the term is multiplied by a constant, then the numerator is that constant).
Indices to the power of another index[edit | edit source]
When a number is raised to the power of an index, then this term is raised to the power of another index, the two indices are multiplied.
It is tempting to think these mechanical rules always work as stated. However, this is not always the case. Consider
but this is actually wrong, because we have failed to account for any possible negative values of . The correct way to "reduce" this is then
Use and manipulation of surds[edit | edit source]
A surd is an irrational root of a whole number such as and . Like indices, laws of use and manipulation of surds is vital knowlegde.
Multiplication and simplification[edit | edit source]
Sometimes you will be asked to simplify your answer, this is done simply by finding a square number, like 4 or 9, that divides into the surd, then bringing it outside the square root.
Surds in fractions[edit | edit source]
Rationalising the denominator[edit | edit source]
Some times you may be given a fraction with a surd as the denominator and be asked to rationalise the denominator. If you are given a fraction with a single surd as a denominator you can simply multiply numerator and denominator by the surd to get rid of it. In effect you are always multiplying by 1.
For a fraction that had a constant plus a surd, for example , you need to multiply numerator and denominator by .
One of the reasons it is useful to rationalise a fraction is because a rationalised expression is easier to evaluate approximately by inspection. Consider the fraction , whose value is not easy to eyeball. However, the equivalent fraction is easier to estimate, as we know that , so . More precisely, we have .
Quadratic functions and their graphs[edit | edit source]
The discriminant of a quadratic function[edit | edit source]
We consider the following quadratic function, where :
We define the discriminant of as the following quantity:
The discriminant allows us to classify the roots or zeroes of . In particular, if we will have two distinct real roots; if we will have no real roots, and if , we will have one (repeated) real root.
Completing the square[edit | edit source]
Simultaneous equations[edit | edit source]
Questions[edit | edit source]
Here are some questions on the above topics to test your knowledge, answers are Here.
Laws of indices[edit | edit source]
So The Laws Of Indices Are Seven Which Are: 1) X^a*X^b=X^a^+^b 2) X^a /X^b=X^a^-^b 3) X^0=1 4) X^-b=1/b 5) X^(a/b)=(b\X)^a 6) (X^a)^b=X^a^b
7) X^1= X Evaluate the following terms.
Surds[edit | edit source]
Express the following in the form .
Rationalise the denominator of the following