A17-SquareRoots.html

High School Modules > Algebra by Gregory A. Moore

Square Roots

We examine the notion of the square root graphically and numerically, then use this understanding to simplify square roots and square root expressions. We also look at fractional exponents for square roots.

[Directions : Execute the Code Resource section first. Although there will be no output immediately, these definitions are used later in this worksheet.]

0. Code

> restart; with(plots): with(RealDomain):

Warning, the name changecoords has been redefined

Warning, these protected names have been redefined and unprotected: Im, Re, ^, arccos, arccosh, arccot, arccoth, arccsc, arccsch, arcsec, arcsech, arcsin, arcsinh, arctan, arctanh, cos, cosh, cot, coth, csc, csch, eval, exp, expand, limit, ln, log, sec, sech, signum, simplify, sin, sinh, solve, sqrt, surd, tan, tanh

>    StandardPlot := proc( number )
local   N, n, A, h, v, i, ps;
N := floor(number):
n := ceil( sqrt(N)):
A := plot( sqrt(x), x = -1..(N+1), tickmarks = [N-1,n-1] ):
h := seq(plot( [[0,k],[N,k]], color = blue), k = 1..n):
v := seq(plot( [[k,0],[k,n]], color = blue), k = 1..N):
for i from 0 to N do
    if( floor( sqrt(i)) = sqrt(i) )
    then ps||i := plottools[disk]([i, sqrt(i)], .25, color=red):
    else ps||i := plottools[disk]([i, sqrt(i)], .15, color=yellow):
    fi:
od:

plots[display]( { A,h,v,seq(ps||i, i = 0..N)}, scaling=constrained );
end proc:

1. Examing A Graph of the Square Root

If we look at the graph the square root on a grid, we see that the curve only passes through certain grid points.

> StandardPlot(10);

In particular, the curve hits the grid intersections at x = 0, 1, 4, 9. Let's look at a large diagram of this sort.

> StandardPlot(27);

We see red dots at x = 0,1, 4, 9, 16, 25. It appears that the square root has integer values only for values
which are perfect squares.

2. Tables

Here is a table of perfect squares. The left column is a number, and the right number its square. You may recognize the numbers in the right column as the same numbers which the square root curve crossed the grid intersections.

> array( [seq( [ k, k^2 ],k = 1..18)]);

What number, when squared, gives 121? The answer to this is the "square root" - the inverse of the square

> sqrt(121);

And sure enough, if we check it by squaring, we get back to the original number.

> 11^2;

What number, when squared, gives 324?

> sqrt(324);

What number, when squared, gives 2500?

> sqrt(2500);

If we write these numbers in the other direction, we have a square root table. This shows that the square root of a number which is a square is just the original number.

> array( [seq( [ k^2 , k],k = 1..13)]);

What about the square roots of other numbers, which are not perfect squares? A square root of a whole number is either another whole number or an irrational number.

> array( [seq( [ k , sqrt(k) = evalf( sqrt(k), 15) ],k = 1..13) ]);