Chapter 1

Find the solutions of the equation $$ x^{2}-2 x+26=0 $$

Express as a polynomial. $$ (3 x-4 y)^{3} $$

Exer. 11-46: Simplify. $$ \left(3 x^{1 / 2}\right)\left(-2 x^{5 / 2}\right) $$

Exer. 33-40: Replace the symbol \(\square\) with either \(=\) or \(\neq\) to make the resulting statement true for all real numbers \(a, b\), \(c\), and \(d\), whenever the expressions are defined. $$ -(a+b) \square-a+b $$

Find the solutions of the equation $$ x^{2}+4 x+13=0 $$

Express as a polynomial. $$ (a+b-c)^{2} $$

Exer. 11-46: Simplify. $$ \left(\frac{-8 x^{3}}{y^{-6}}\right)^{2 / 3} $$

Exer. 41-42: Approximate the real-number expression to four decimal places. (a) \(\left|3.2^{2}-\sqrt{3.15}\right|\) (b) \(\sqrt{(15.6-1.5)^{2}+(4.3-5.4)^{2}}\)

Find the solutions of the equation $$ x^{2}+8 x+17=0 $$

Express as a polynomial. $$ \left(x^{2}+x+1\right)^{2} $$

Exer. 11-46: Simplify. $$ \left(\frac{-y^{3 / 2}}{y^{-1 / 3}}\right)^{3} $$

Exer. 41-42: Approximate the real-number expression to four decimal places. (a) \(\frac{3.42-1.29}{5.83+2.64}\) (b) \(\pi^{3}\)

Find the solutions of the equation $$ x^{2}-5 x+20=0 $$

Express as a polynomial. $$ (2 x+y-3 z)^{2} $$

Exer. 11-46: Simplify. $$ \left(\frac{x^{6}}{9 y^{-4}}\right)^{-1 / 2} $$

Exer. 43-44: Approximate the real-number expression. Express the answer in scientific notation accurate to four significant figures. (a) \(\frac{1.2 \times 10^{3}}{3.1 \times 10^{2}+1.52 \times 10^{3}}\) (b) \(\left(1.23 \times 10^{-4}\right)+\sqrt{4.5 \times 10^{3}}\)

Find the solutions of the equation $$ x^{2}+3 x+6=0 $$

Express as a polynomial. $$ (x-2 y+3 z)^{2} $$

Exer. 11-46: Simplify. $$ \left(\frac{c^{-4}}{16 d^{8}}\right)^{3 / 4} $$

Exer. 43-44: Approximate the real-number expression. Express the answer in scientific notation accurate to four significant figures. (a) \(\sqrt{\left|3.45-1.2 \times 10^{4}\right|+10^{5}}\) (b) \(\left(1.791 \times 10^{2}\right) \times\left(9.84 \times 10^{3}\right)\)

Find the solutions of the equation $$ 4 x^{2}+x+3=0 $$

Factor the polynomial. $$ r s+4 s t $$

Exer. 11-46: Simplify. $$ \frac{\left(x^{6} y^{3}\right)^{-1 / 3}}{\left(x^{4} y^{2}\right)^{-1 / 2}} $$

Factor the polynomial. $$ 4 u^{2}-2 u v $$

Exer. 11-46: Simplify. $$ a^{4 / 3} a^{-3 / 2} a^{1 / 6} $$

A circle of radius 1 rolls along a coordinate line in the positive direction, as shown in the figure. If point \(P\) is initially at the origin, find the coordinate of \(P\) after one, two, and ten complete revolutions.

Find the solutions of the equation $$ x^{3}+125=0 $$

Factor the polynomial. $$ 3 a^{2} b^{2}-6 a^{2} b $$

Exer. 47-52: Rewrite the expression using rational exponents. $$ \sqrt[4]{x^{3}} $$

Factor the polynomial. $$ 10 x y+15 x y^{2} $$

Exer. 47-52: Rewrite the expression using rational exponents. $$ \sqrt[3]{x^{5}} $$

Rational approximations to square roots can be found using a formula discovered by the ancient Babylonians. Let \(x_{1}\) be the first rational approximation for \(\sqrt{n}\). If we let $$ x_{2}=\frac{1}{2}\left(x_{1}+\frac{n}{x_{1}}\right) $$ then \(x_{2}\) will be a better approximation for \(\sqrt{n}\), and we can repeat the computation with \(x_{2}\) replacing \(x_{1}\). Starting with \(x_{1}=\frac{3}{2}\), find the next two rational approximations for \(\sqrt{2}\).

Find the solutions of the equation $$ 27 x^{3}=(x+5)^{3} $$

Factor the polynomial. $$ 3 x^{2} y^{3}-9 x^{3} y^{2} $$

Exer. 47-52: Rewrite the expression using rational exponents. $$ \sqrt[3]{(a+b)^{2}} $$

Exer. 49-50: Express the number in scientific form. (a) 427,000 (b) \(0.000000098\) (c) \(810,000,000\)

Factor the polynomial. $$ 16 x^{5} y^{2}+8 x^{3} y^{3} $$

Exer. 47-52: Rewrite the expression using rational exponents. $$ \sqrt{a+\sqrt{b}} $$

Exer. 49-50: Express the number in scientific form. (a) 427,000 (b) \(0.000000098\) (c) \(810,000,000\)

Factor the polynomial. $$ 15 x^{3} y^{5}-25 x^{4} y^{2}+10 x^{6} y^{4} $$

Exer. 47-52: Rewrite the expression using rational exponents. $$ \sqrt{x^{2}+y^{2}} $$

Exer. 51-52: Express the number in decimal form. (a) \(8.3 \times 10^{5}\) (b) \(2.9 \times 10^{-12}\) (c) \(5.63 \times 10^{8}\)

Factor the polynomial. $$ 121 r^{3} s^{4}+77 r^{2} s^{4}-55 r^{4} s^{3} $$

Exer. 47-52: Rewrite the expression using rational exponents. $$ \sqrt[3]{r^{3}-s^{3}} $$

Exer. 51-52: Express the number in decimal form. (a) \(2.3 \times 10^{7}\) (b) \(7.01 \times 10^{-9}\) (c) \(1.23 \times 10^{10}\)

Factor the polynomial. $$ 8 x^{2}-53 x-21 $$

Exer. 53-56: Rewrite the expression using a radical. (a) \(4 x^{3 / 2}\) (b) \((4 x)^{3 / 2}\)

Mass of a hydrogen atom The mass of a hydrogen atom is approximately $$ 0.0000000000000000000000017 \text { gram. } $$ Express this number in scientific form.

Factor the polynomial. $$ 7 x^{2}+10 x-8 $$

Exer. 53-56: Rewrite the expression using a radical. (a) \(4+x^{3 / 2}\) (b) \((4+x)^{3 / 2}\)