Chapter 1

Factor the polynomial. $$ 64 x^{2}-36 y^{2} $$

Exer. 57-80: Simplify the expression, and rationalize the denominator when appropriate. $$ \sqrt[6]{\left(2 u^{-3} v^{4}\right)^{6}} $$

Factor the polynomial. $$ 64 x^{3}+27 $$

Exer. 57-80: Simplify the expression, and rationalize the denominator when appropriate. $$ \sqrt[5]{\frac{8 x^{3}}{y^{4}}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}} $$

Factor the polynomial. $$ 125 x^{3}-8 $$

Exer. 57-80: Simplify the expression, and rationalize the denominator when appropriate. $$ \sqrt{5 x y^{7}} \sqrt{10 x^{3} y^{3}} $$

Factor the polynomial. $$ 64 x^{3}-y^{6} $$

Exer. 57-80: Simplify the expression, and rationalize the denominator when appropriate. $$ \sqrt[3]{3 t^{4} v^{2}} \sqrt[3]{-9 t^{-1} v^{4}} $$

Factor the polynomial. $$ 216 x^{9}+125 y^{3} $$

Exer. 57-80: Simplify the expression, and rationalize the denominator when appropriate. $$ \sqrt[3]{(2 r-s)^{3}} $$

Factor the polynomial. $$ 343 x^{3}+y^{9} $$

Exer. 81-84: Simplify the expression, assuming \(x\) and \(y\) may be negative. $$ \sqrt{x^{6} y^{4}} $$

Factor the polynomial. $$ x^{6}-27 y^{3} $$

Exer. 81-84: Simplify the expression, assuming \(x\) and \(y\) may be negative. $$ \sqrt{x^{4} y^{10}} $$

Factor the polynomial. $$ 125-27 x^{3} $$

Exer. 81-84: Simplify the expression, assuming \(x\) and \(y\) may be negative. $$ \sqrt[4]{x^{8}(y-1)^{12}} $$

Factor the polynomial. $$ x^{3}+64 $$

Exer. 81-84: Simplify the expression, assuming \(x\) and \(y\) may be negative. $$ \sqrt[4]{(x+2)^{12} y^{4}} $$

Factor the polynomial. $$ 2 a x-6 b x+a y-3 b y $$

Exer. 85-90: Replace the symbol \(\square\) with either \(=\) or \(\neq\) to make the resulting statement true, whenever the expression has meaning. Give a reason for your answer. $$ \left(a^{r}\right)^{2} \square a^{\left(r^{2}\right)} $$

Factor the polynomial. $$ 2 a y^{2}-a x y+6 x y-3 x^{2} $$

Exer. 85-90: Replace the symbol \(\square\) with either \(=\) or \(\neq\) to make the resulting statement true, whenever the expression has meaning. Give a reason for your answer. $$ \left(a^{2}+1\right)^{1 / 2} \square a+1 $$

Factor the polynomial. $$ 3 x^{3}+3 x^{2}-27 x-27 $$

Exer. 85-90: Replace the symbol \(\square\) with either \(=\) or \(\neq\) to make the resulting statement true, whenever the expression has meaning. Give a reason for your answer. $$ a^{x} b^{y} \square(a b)^{x y} $$

Factor the polynomial. $$ 5 x^{3}+10 x^{2}-20 x-40 $$

Factor the polynomial. $$ x^{4}+2 x^{3}-x-2 $$

Exer. 85-90: Replace the symbol \(\square\) with either \(=\) or \(\neq\) to make the resulting statement true, whenever the expression has meaning. Give a reason for your answer. $$ \sqrt[n]{\frac{1}{c}} \square \frac{1}{\sqrt[n]{c}} $$

Factor the polynomial. $$ x^{4}-3 x^{3}+8 x-24 $$

Exer. 85-90: Replace the symbol \(\square\) with either \(=\) or \(\neq\) to make the resulting statement true, whenever the expression has meaning. Give a reason for your answer. $$ a^{1 / k} \square \frac{1}{a^{k}} $$

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

Exer. 91-92: In evaluating negative numbers raised to fractional powers, it may be necessary to evaluate the root and integer power separately. For example, \((-3)^{2 / 5}\) can be evaluated successfully as \(\left[(-3)^{1 / 5}\right]^{2}\) or \(\left[(-3)^{2}\right]^{1 / 5}\), whereas an error message might otherwise appear. Approximate the realnumber expression to four decimal places. $$ \text { (a) }(-3)^{2 / 5} $$

Factor the polynomial. $$ 6 w^{8}+17 w^{4}+12 $$

Exer. 91-92: In evaluating negative numbers raised to fractional powers, it may be necessary to evaluate the root and integer power separately. For example, \((-3)^{2 / 5}\) can be evaluated successfully as \(\left[(-3)^{1 / 5}\right]^{2}\) or \(\left[(-3)^{2}\right]^{1 / 5}\), whereas an error message might otherwise appear. Approximate the realnumber expression to four decimal places. (a) \((-1.2)^{3 / 7}\) (b) \((-5.08)^{7 / 3}\)

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

Exer. 93-94: Approximate the real-number expression to four decimal places. (a) \(\sqrt{\pi+1}\) (b) \(\sqrt[3]{15.1}+5^{1 / 4}\)

Factor the polynomial. $$ x^{8}-16 $$

Exer. 93-94: Approximate the real-number expression to four decimal places. (a) \((2.6-1.9)^{-2}\) (b) \(5^{\sqrt{7}}\)

Factor the polynomial. $$ x^{2}+4 x+4-9 y^{2} $$

One of the oldest banks in the United States is the Bank of America, founded in 1812. If \(\$ 200\) had been deposited at that time into an account that paid \(4 \%\) annual interest, then 180 years later the amount would have grown to \(200(1.04)^{180}\) dollars. Approximate this amount to the nearest cent.

Factor the polynomial. $$ x^{2}-4 y^{2}-6 x+9 $$

Viewing distance On a clear day, the distance \(d\) (in miles) that can be seen from the top of a tall building of height \(h\) (in feet) can be approximated by \(d=1.2 \sqrt{h}\). Approximate the distance that can be seen from the top of the Chicago Sears Tower, which is 1454 feet tall.

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

The length-weight relationship for Pacific halibut can be approximated by the formula \(L=0.46 \sqrt[3]{W}\), where \(W\) is in kilograms and \(L\) is in meters. The largest documented halibut weighed 230 kilograms. Estimate its length.

Factor the polynomial. $$ y^{2}+9-6 y-4 x^{2} $$

The length-weight relationship for the sei whale can be approximated by \(W=0.0016 L^{2.43}\), where \(W\) is in tons and \(L\) is in feet. Estimate the weight of a whale that is 25 feet long.

Factor the polynomial. $$ y^{6}+7 y^{3}-8 $$

O'Carroll's formula is used to handicap weight lifters. If a lifter who weighs \(b\) kilograms lifts \(w\) kilograms of weight, then the handicapped weight \(W\) is given by $$ W=\frac{w}{\sqrt[3]{b-35}} $$ Suppose two lifters weighing 75 kilograms and 120 kilograms lift weights of 180 kilograms and 250 kilograms, respectively. Use O'Carroll's formula to determine the superior weight lifter.

Factor the polynomial. $$ 8 c^{6}+19 c^{3}-27 $$

Body surface area $$ S=(0.1091) w^{0.425} h^{0.725}, $$ where height \(h\) is in inches and weight \(w\) is in pounds. (a) Estimate \(S\) for a person 6 feet tall weighing 175 pounds. (b) If a person is 5 feet 6 inches tall, what effect does a \(10 \%\) increase in weight have on \(S\) ?

Factor the polynomial. $$ x^{16}-1 $$