Chapter 1: Fundamentals

Rationalize the numerator.

$\sqrt{x^2+1}-x$

Factor the expression completely. Begin by factoring out the lowest power of each common factor.

${\left(x^2+1\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+2{\left(x^2+1\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

Let a, b, and c be real numbers with $a>0,b<0$, and $c<0$. Determine the sign of each expression.

1. $b^5$
2. $b^{10}$
3. $a{b}^2{c}^3$
4. ${(b-a)}^3$
5. ${(b-a)}^4$
6. $\frac{a^3{c}^3}{b^6{c}^6}$

We prove Property 5 of absolute values, the Triangle Inequality:

$|x+y|\le \left|x\right|+\left|y\right|$

(a) Verify that the Triangle Inequality holds for $x=2,y=3$, for$x=-2,y=-3$ , and for$x=-2,y=3$ .

(b) Prove that the Triangle Inequality is true for all real numbers x and y.

Find an equation of the circle that satisfies the given conditions.

Centre $(7,-3)$; tangent to thex-axis

Sketch the graph of region of the set $\left\{\left(x,y\right)\right\}\mathbf{|}\mathbf{-}\mathbf{4}\mathbf{<}\mathbf{x}\mathbf{<}\mathbf{4}\mathbf{and}\mathbf{-}\mathbf{2}\mathbf{<}\mathbf{y}\mathbf{<}\mathbf{2}$.

Factor the expression completely. Begin by factoring out the lowest power of each common factor.

$x^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\left(x+1\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\left(x+1\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

Find an equation of the circle that satisfies the given conditions.

The circle lies in the first quadrant tangent to both x-and y-axes; radius5

Find the real solutions of the equation $\sqrt{3+x}=\sqrt{x^2+1}$.

Rationalize the numerator.

$\sqrt{x+1}-\sqrt{x}$