# Algebra/Powers

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## Laws of indices of all rational exponents

An index is of the form $a^{x}$ , and the laws on how to manipulate them is vital knowledge.

### Power of one and zero

Any nonzero base raised to the power of one is simply the base.

For example,

$a^{1}=a$ $8^{1}=8$ Any nonzero base raised to the power of zero is one.

For example,

$a^{0}=1$ $5^{0}=1$ $0^{0}$ has no meaning.

### Multiplication and division

When two indices are multiplied, as long as the bases are equal, the two indices are simply added together.

For example,

$a^{x}\times a^{y}=a^{x+y}$ $2^{2}\times 2^{3}=2^{2+3}=2^{5}=32$ For division the opposite is true, as long as the bases are equal, the indices are subtracted.

For example,

$a^{x}\div a^{y}=a^{x-y}$ $2^{5}\div 2^{2}=2^{5-2}=2^{3}=8$ ### Fractional indices

The denominator of a fractional index is the root that the base must be taken to.

For example,

$a^{\frac {1}{2}}={\sqrt {a}}$ $a^{\frac {1}{3}}={\sqrt[{3}]{a}}$ The numerator of a fractional index is the power the base must be raised to.

For example,

$a^{\frac {2}{3}}={\sqrt[{3}]{a^{2}}}$ $8^{\frac {2}{3}}={\sqrt[{3}]{8^{2}}}=2^{2}=4$ ### Negative indices

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).

For example,

$a^{-1}={\frac {1}{a}}$ $a^{-3}={\frac {1}{a^{3}}}$ $2(a^{-1})={\frac {2}{a}}$ ### Indices to the power of another index

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.

For example,

$\left(a^{x}\right)^{y}=a^{x\times y}=a^{xy}$ $\left(a^{2}\right)^{3}=a^{2\times 3}=a^{6}$ $\left({\sqrt {a}}\right)^{4}=a^{{\frac {1}{2}}\times 4}=a^{2}$ It is tempting to think these mechanical rules always work as stated. However, this is not always the case. Consider

${\sqrt {a^{2}}}=(a^{2})^{1/2}=a^{2\times {\frac {1}{2}}}=a$ but this is actually wrong, because we have failed to account for any possible negative values of $a$ . The correct way to "reduce" this is then

${\sqrt {a^{2}}}=|a|$ ## Use and manipulation of surds

A surd is an irrational root of a whole number such as ${\sqrt {2}}$ and ${\sqrt {5}}$ . Like indices, laws of use and manipulation of surds is vital knowlegde.

### Multiplication and simplification

${\sqrt {a}}\times {\sqrt {b}}={\sqrt {a\times b}}$ ${\sqrt {2}}\times {\sqrt {8}}={\sqrt {16}}=4$ 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.

For example,

${\sqrt {50}}={\sqrt {25\times 2}}={\sqrt {25}}\times {\sqrt {2}}=5{\sqrt {2}}$ ### Surds in fractions

${\frac {\sqrt {a}}{\sqrt {b}}}={\sqrt {\frac {a}{b}}}$ ${\sqrt {\frac {5}{4}}}={\frac {\sqrt {5}}{2}}$ #### Rationalising the denominator

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 example,

${\frac {3}{\sqrt {2}}}\times \left({\frac {\sqrt {2}}{\sqrt {2}}}\right)={\frac {3{\sqrt {2}}}{2}}$ For a fraction that had a constant plus a surd, for example $a+{\sqrt {b}}$ , you need to multiply numerator and denominator by $a-{\sqrt {b}}$ .

For example,

${\frac {3}{3+{\sqrt {2}}}}\times \left({\frac {3-{\sqrt {2}}}{3-{\sqrt {2}}}}\right)={\frac {3\left(3-{\sqrt {2}}\right)}{9-3{\sqrt {2}}+3{\sqrt {2}}-2}}={\frac {9-3{\sqrt {2}}}{7}}$ 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 $1/{\sqrt {2}}$ , whose value is not easy to eyeball. However, the equivalent fraction ${\sqrt {2}}/2$ is easier to estimate, as we know that ${\sqrt {2}}\approx 1.4$ , so ${\sqrt {2}}/2\approx 0.7$ . More precisely, we have $1/{\sqrt {2}}=0.70710678118\ldots$ .

## The discriminant of a quadratic function

We consider the following quadratic function, where $A\neq 0$ :

$f(x)=Ax^{2}+Bx+C.$ We define the discriminant of $f$ as the following quantity:

${\text{Discriminant of }}f={\text{Disc}}(f)=B^{2}-4AC.$ The discriminant allows us to classify the roots or zeroes of $f$ . In particular, if ${\text{Disc}}(f)>0$ we will have two distinct real roots; if ${\text{Disc}}(f)<0$ we will have no real roots, and if ${\text{Disc}}(f)=0$ , we will have one (repeated) real root.

## Questions

Here are some questions on the above topics to test your knowledge, answers are Here.

### Laws of indices

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.

1. $2^{2}\times 2^{3}$ 2. $27^{\frac {2}{3}}$ 3. $4^{-{\frac {3}{2}}}$ 4. $(-5)^{-3}$ 5. $9^{-0.5}$ 6. $\left(x^{3}\right)^{\frac {2}{3}}$ (Hortharn (discuss • contribs) 00:28, 18 April 2018 (UTC))


### Surds

Express the following in the form $a{\sqrt {b}}$ .

1. ${\sqrt {45}}$ 2. ${\sqrt {12}}$ 3. ${\sqrt {32}}$ Rationalise the denominator of the following

1. ${\frac {4}{3-{\sqrt {5}}}}$ 2. ${\frac {11}{3+{\sqrt {11}}}}$ 3. ${\frac {3-{\sqrt {2}}}{4-{\sqrt {5}}}}$ 