Chapter 1 Prerequisites
Chapter 1 Review Exercises
Chapter Review Exercises
Real Numbers: Algebra Essentials
For the following exercises, perform the given operations.
- [latex]{\left(5-3\cdot 2\right)}^{2}-6[/latex]
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[latex]-5[/latex]
- [latex]64÷\left(2\cdot 8\right)+14÷7[/latex]
- [latex]2\cdot {5}^{2}+6÷2[/latex]
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53
For the following exercises, solve the equation.
- [latex]5x+9=-11[/latex]
- [latex]2y+{4}^{2}=64[/latex]
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[latex]y=24[/latex]
For the following exercises, simplify the expression.
- [latex]9\left(y+2\right)÷3\cdot 2+1[/latex]
- [latex]3m\left(4+7\right)-m[/latex]
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[latex]32m[/latex]
For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.
- 11
- 0
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whole
- [latex]\frac{5}{6}[/latex]
- [latex]\sqrt{11}[/latex]
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irrational
Exponents and Scientific Notation
For the following exercises, simplify the expression.
- [latex]{2}^{2}\cdot {2}^{4}[/latex]
- [latex]\frac{{4}^{5}}{{4}^{3}}[/latex]
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[latex]16[/latex]
- [latex]{\left(\frac{{a}^{2}}{{b}^{3}}\right)}^{4}[/latex]
- [latex]\frac{6{a}^{2}\cdot {a}^{0}}{2{a}^{-4}}[/latex]
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[latex]{a}^{6}[/latex]
- [latex]\frac{{\left(xy\right)}^{4}}{{y}^{3}}\cdot \frac{2}{{x}^{5}}[/latex]
- [latex]\frac{{4}^{-2}{x}^{3}{y}^{-3}}{2{x}^{0}}[/latex]
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[latex]\frac{{x}^{3}}{32{y}^{3}}[/latex]
- [latex]{\left(\frac{2{x}^{2}}{y}\right)}^{-2}[/latex]
- [latex]\left(\frac{16{a}^{3}}{{b}^{2}}\right){\left(4a{b}^{-1}\right)}^{-2}[/latex]
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[latex]a[/latex]
- Write the number in standard notation:[latex]\,2.1314\,×\,{10}^{-6}[/latex]
- Write the number in scientific notation: 16,340,000
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[latex]1.634\,×\,{10}^{7}[/latex]
Radicals and Rational Expressions
For the following exercises, find the principal square root.
- [latex]\sqrt{121}[/latex]
- [latex]\sqrt{196}[/latex]
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14
- [latex]\sqrt{361}[/latex]
- [latex]\sqrt{75}[/latex]
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[latex]5\sqrt{3}[/latex]
- [latex]\sqrt{162}[/latex]
- [latex]\sqrt{\frac{32}{25}}[/latex]
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[latex]\frac{4\sqrt{2}}{5}[/latex]
- [latex]\sqrt{\frac{80}{81}}[/latex]
- [latex]\sqrt{\frac{49}{1250}}[/latex]
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[latex]\frac{7\sqrt{2}}{50}[/latex]
- [latex]\frac{2}{4+\sqrt{2}}[/latex]
- [latex]4\sqrt{3}+6\sqrt{3}[/latex]
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[latex]10\sqrt{3}[/latex]
- [latex]12\sqrt{5}-13\sqrt{5}[/latex]
- [latex]\sqrt[5]{-243}[/latex]
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[latex]-3[/latex]
- [latex]\frac{\sqrt[3]{250}}{\sqrt[3]{-8}}[/latex]
Polynomials
For the following exercises, perform the given operations and simplify.
- [latex]\left(3{x}^{3}+2x-1\right)+\left(4{x}^{2}-2x+7\right)[/latex]
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[latex]3{x}^{3}+4{x}^{2}+6[/latex]
- [latex]\left(2y+1\right)-\left(2{y}^{2}-2y-5\right)[/latex]
- [latex]\left(2{x}^{2}+3x-6\right)+\left(3{x}^{2}-4x+9\right)[/latex]
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[latex]5{x}^{2}-x+3[/latex]
- [latex]\left(6{a}^{2}+3a+10\right)-\left(6{a}^{2}-3a+5\right)[/latex]
- [latex]\left(k+3\right)\left(k-6\right)[/latex]
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[latex]{k}^{2}-3k-18[/latex]
- [latex]\left(2h+1\right)\left(3h-2\right)[/latex]
- [latex]\left(x+1\right)\left({x}^{2}+1\right)[/latex]
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[latex]{x}^{3}+{x}^{2}+x+1[/latex]
- [latex]\left(m-2\right)\left({m}^{2}+2m-3\right)[/latex]
- [latex]\left(a+2b\right)\left(3a-b\right)[/latex]
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[latex]3{a}^{2}+5ab-2{b}^{2}[/latex]
- [latex]\left(x+y\right)\left(x-y\right)[/latex]
Factoring Polynomials
For the following exercises, find the greatest common factor.
- [latex]81p+9pq-27{p}^{2}{q}^{2}[/latex]
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[latex]9p[/latex]
- [latex]12{x}^{2}y+4x{y}^{2}-18xy[/latex]
- [latex]88{a}^{3}b+4{a}^{2}b-144{a}^{2}[/latex]
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[latex]4{a}^{2}[/latex]
For the following exercises, factor the polynomial.
- [latex]2{x}^{2}-9x-18[/latex]
- [latex]8{a}^{2}+30a-27[/latex]
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[latex]\left(4a-3\right)\left(2a+9\right)[/latex]
- [latex]{d}^{2}-5d-66[/latex]
- [latex]{x}^{2}+10x+25[/latex]
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[latex]{\left(x+5\right)}^{2}[/latex]
- [latex]{y}^{2}-6y+9[/latex]
- [latex]4{h}^{2}-12hk+9{k}^{2}[/latex]
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[latex]{\left(2h-3k\right)}^{2}[/latex]
- [latex]361{x}^{2}-121[/latex]
- [latex]{p}^{3}+216[/latex]
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[latex]\left(p+6\right)\left({p}^{2}-6p+36\right)[/latex]
- [latex]8{x}^{3}-125[/latex]
- [latex]64{q}^{3}-27{p}^{3}[/latex]
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[latex]\left(4q-3p\right)\left(16{q}^{2}+12pq+9{p}^{2}\right)[/latex]
- [latex]4x{\left(x-1\right)}^{-\frac{1}{4}}+3{\left(x-1\right)}^{\frac{3}{4}}[/latex]
- [latex]3p{\left(p+3\right)}^{\frac{1}{3}}-8{\left(p+3\right)}^{\frac{4}{3}}[/latex]
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[latex]{\left(p+3\right)}^{\frac{1}{3}}\left(-5p-24\right)[/latex]
- [latex]4r{\left(2r-1\right)}^{-\frac{2}{3}}-5{\left(2r-1\right)}^{\frac{1}{3}}[/latex]
Rational Expressions
For the following exercises, simplify the expression.
- [latex]\frac{{x}^{2}-x-12}{{x}^{2}-8x+16}[/latex]
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[latex]\frac{x+3}{x-4}[/latex]
- [latex]\frac{4{y}^{2}-25}{4{y}^{2}-20y+25}[/latex]
- [latex]\frac{2{a}^{2}-a-3}{2{a}^{2}-6a-8}\cdot \frac{5{a}^{2}-19a-4}{10{a}^{2}-13a-3}[/latex]
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[latex]\frac{1}{2}[/latex]
- [latex]\frac{d-4}{{d}^{2}-9}\cdot \frac{d-3}{{d}^{2}-16}[/latex]
- [latex]\frac{{m}^{2}+5m+6}{2{m}^{2}-5m-3}÷\frac{2{m}^{2}+3m-9}{4{m}^{2}-4m-3}[/latex]
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[latex]\frac{m+2}{m-3}[/latex]
- [latex]\frac{4{d}^{2}-7d-2}{6{d}^{2}-17d+10}÷\frac{8{d}^{2}+6d+1}{6{d}^{2}+7d-10}[/latex]
- [latex]\frac{10}{x}+\frac{6}{y}[/latex]
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[latex]\frac{6x+10y}{xy}[/latex]
- [latex]\frac{12}{{a}^{2}+2a+1}-\frac{3}{{a}^{2}-1}[/latex]
- [latex]\frac{\frac{1}{d}+\frac{2}{c}}{\frac{6c+12d}{dc}}[/latex]
Show Solution
[latex]\frac{1}{6}[/latex]
- [latex]\frac{\frac{3}{x}-\frac{7}{y}}{\frac{2}{x}}[/latex]