Powers and radicals

Year 9

Operations with real numbers

Natural powers of real numbers

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Powers of real numbers extend to rational exponents, positive or negative, and to real exponents as well.

Fundamental identities

For any $x,y,z,t,a,b,c,d\in R$ and $n\in Ν$, we have:

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Definition. Let A be a symmetric set and $f:A\to B$ a numerical function. The function f is called even if for $(\forall)x\in A$ we have $f(-x)=f(x)$. The function f is called odd if for $(\forall)x\in A$ we have $f(-x)=-f(x)$

Theorem. If n is even, the power function is an even function and its graph is symmetric about the OY axis. If n is odd, the power function is an odd function and its graph is symmetric about the origin.

Graphs of the power functions

1. if n=2 (parabola);

2. if n=3 (cubic parabola);

3. if n=-1 (hyperbola);

4. if n=-1 ;

A comparison between the graphs of f(x)=x² and g(x)=x³: the left-hand graph shows the two functions on the interval [0,4], and the right-hand graph shows the same functions on the interval [0,1.6] after zooming in.

Radicals

Properties.

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Graph of the radical function and the graph of the inverse function

1. for n=2 $\Rightarrow y:[0,\infty)\to[0,\infty),$ ; the function y(x) is strictly increasing and bijective on the interval $[0,\infty)$. The graph of the inverse function is also shown. The two graphs are symmetric about the first bisector, the line $y(x)=x$, drawn dotted.

2. for n=3 $\Rightarrow y:\mathbb{R}\to \mathbb{R},$ ;

The modulus of a real number.

Properties:

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Graph of the function , $(\forall)x\in \mathbb{R}$

Means

The arithmetic mean of the numbers a₁,a₂ . . a_n is .

The harmonic mean of a₁,a₂ . . a_n is ,

The geometric mean of a₁,a₂ . . a_n is ,

The inequality of means. Let and ; we have:

$\le$$\le$

The integer part of a number a is the largest integer smaller than the number a, written . The fractional part of a number a is the difference between the number and its integer part, written . We have , .

For , we have

The signum (sign) function