Chapter 1

irrational number. The square root of two does not terminate, and it does not repeat a pattern. It cannot be written as a quotient of two integers, so it is irrational.

The Associative Properties state that the sum or product of multiple numbers can be grouped differently without affecting the result. This is because the same operation is performed (either addition or subtraction), so the terms can be re-ordered.

$\mathrm{-6}$

$\mathrm{-2}$

$\mathrm{-9}$

9

-2

4

0

9

25

$\mathrm{-6}$

17

4

$14$

$\mathrm{-66}$

$\mathrm{–12}$

$\mathrm{–44}$

$\mathrm{–2}$

$\mathrm{-14}y-11$

$\mathrm{-4}b+1$

$43z-3$

$9y+45$

$\mathrm{-6}b+6$

$\frac{16x}{3}$

$9x$

$\frac{1}{2}\left(40-10\right)+5$

irrational number

$g+400-2\left(600\right)=1200$

inverse property of addition

68.4

true

irrational

rational

No, the two expressions are not the same. An exponent tells how many times you multiply the base. So $2^3$ is the same as $2\times 2\times 2,$ which is 8. $3^2$ is the same as $3\times 3,$ which is 9.

It is a method of writing very small and very large numbers.

81

243

$\frac{1}{16}$

$\frac{1}{11}$

1

$4^9$

${12}^{40}$

$\frac{1}{7^9}$

$3.14\times {10}^{-5}$

16,000,000,000

$a^4$

$b^6{c}^8$

$a{b}^2{d}^3$

$m^4$

$\frac{q^5}{p^6}$

$\frac{y^{21}}{x^{14}}$

$25$

$72a^2$

$\frac{c^3}{b^9}$

$\frac{y}{81z^6}$

0.00135 m

$1.0995\times {10}^{12}$

0.00000000003397 in.

12,230,590,464 $m^{66}$

$\frac{a^{14}}{1296}$

$\frac{n}{a^{9}c}$

$\frac{1}{a^6{b}^6{c}^6}$

0.000000000000000000000000000000000662606957

When there is no index, it is assumed to be 2 or the square root. The expression would only be equal to the radicand if the index were 1.

The principal square root is the nonnegative root of the number.

16

10

14

$7\sqrt{2}$

$\frac{9\sqrt{5}}{5}$

25

$\sqrt{2}$

$2\sqrt{6}$

$5\sqrt{6}$

$6\sqrt{35}$

$\frac{2}{15}$

$\frac{6\sqrt{10}}{19}$

$-\frac{1+\sqrt{17}}{2}$

$7\sqrt[3]{2}$

$15\sqrt{5}$

$20x^2$

$7\sqrt{p}$

$17m^2\sqrt{m}$

$2b\sqrt{a}$

$\frac{15x}{7}$

$5y^4\sqrt{2}$

$\frac{4\sqrt{7d}}{7d}$

$\frac{2\sqrt{2}+2\sqrt{6x}}{1\mathrm{-3}x}$

$-w\sqrt{2w}$

$\frac{3\sqrt{x}-\sqrt{3x}}{2}$

$5n^5\sqrt{5}$

$\frac{9\sqrt{m}}{19m}$

$\frac{2}{3d}$

$\frac{3\sqrt[4]{2x^2}}{2}$

$6z\sqrt[3]{2}$

500 feet

$\frac{\mathrm{-5}\sqrt{2}\mathrm{-6}}{7}$

$\frac{\sqrt{mnc}}{a^{9}cmn}$

$\frac{2x+1}{2\sqrt{2}}$

$\frac{\sqrt{3}}{3}$

The statement is true. In standard form, the polynomial with the highest value exponent is placed first and is the leading term. The degree of a polynomial is the value of the highest exponent, which in standard form is also the exponent of the leading term.

Use the distributive property, multiply, combine like terms, and simplify.

2

8

2

$4x^2+3x+19$

$3w^2+30w+21$

$11b^4\mathrm{-9}{b}^3+12b^2\mathrm{-7}b+8$

$24x^2\mathrm{-4}x\mathrm{-8}$

$24b^4\mathrm{-48}{b}^2+24$

$99v^2\mathrm{-202}v+99$

$8n^3\mathrm{-4}{n}^2+72n\mathrm{-36}$

$9y^2\mathrm{-42}y+49$

$16p^2+72p+81$

$9y^2\mathrm{-36}y+36$

$16c^2\mathrm{-1}$

$225n^2\mathrm{-36}$

$\mathrm{-16}{m}^2+16$

$121q^2\mathrm{-100}$

$16t^4+4t^3\mathrm{-32}{t}^2-t+7$

$y^3\mathrm{-6}{y}^2-y+18$

$3p^3-p^2\mathrm{-12}p+10$

$a^2-b^2$

$16t^2\mathrm{-40}tu+25u^2$

$4t^2+x^2+4t\mathrm{-5}tx-x$

$24r^2+22rd\mathrm{-7}{d}^2$

$32x^2\mathrm{-4}x\mathrm{-3}$ m²

$32t^3-100t^2+40t+38$

$a^4+4a^{3}c\mathrm{-16}a{c}^3\mathrm{-16}{c}^4$

The terms of a polynomial do not have to have a common factor for the entire polynomial to be factorable. For example, $4x^2$ and $\mathrm{-9}{y}^2$ don’t have a common factor, but the whole polynomial is still factorable: $4x^2\mathrm{-9}{y}^2=\left(2x+3y\right)\left(2x\mathrm{-3}y\right).$

Divide the $x$ term into the sum of two terms, factor each portion of the expression separately, and then factor out the GCF of the entire expression.

$7m$

$10m^3$

$y$

$\left(2a\mathrm{-3}\right)\left(a+6\right)$

$\left(3n\mathrm{-11}\right)\left(2n+1\right)$

$\left(p+1\right)\left(2p\mathrm{-7}\right)$

$\left(5h+3\right)\left(2h\mathrm{-3}\right)$

$\left(9d\mathrm{-1}\right)\left(d\mathrm{-8}\right)$

$\left(12t+13\right)\left(t\mathrm{-1}\right)$

$(4x+10)(4x-10)$

$(11p+13)(11p-13)$

$(19d+9)(19d-9)$

$(12b+5c)(12b-5c)$

${\left(7n+12\right)}^2$

${\left(15y+4\right)}^2$

${(5p-12)}^2$

$(x+6)(x^2-6x+36)$

$(5a+7)(25a^2-35a+49)$

$(4x-5)(16x^2+20x+25)$

$(5r+12s)(25r^2-60rs+144s^2)$

${\left(2c+3\right)}^{-\frac{1}{4}}\left(\mathrm{-7}c-15\right)$

${\left(x+2\right)}^{-\frac{2}{5}}\left(19x+10\right)$

${\left(2z-9\right)}^{-\frac{3}{2}}\left(27z-99\right)$

$\left(14x\mathrm{-3}\right)\left(7x+9\right)$

$\left(3x+5\right)\left(3x\mathrm{-5}\right)$

${(2x+5)}^2{(2x-5)}^2$

$(4z^2+49a^2)(2z+7a)(2z-7a)$

$\frac{1}{\left(4x+9\right)\left(4x\mathrm{-9}\right)\left(2x+3\right)}$

You can factor the numerator and denominator to see if any of the terms can cancel one another out.

True. Multiplication and division do not require finding the LCD because the denominators can be combined through those operations, whereas addition and subtraction require like terms.

$\frac{y+5}{y+6}$

$3b+3$

$\frac{x+4}{2x+2}$

$\frac{a+3}{a-3}$

$\frac{3n-8}{7n-3}$

$\frac{c-6}{c+6}$

$1$

$\frac{d^2-25}{25d^2-1}$

$\frac{t+5}{t+3}$

$\frac{6x-5}{6x+5}$

$\frac{p+6}{4p+3}$

$\frac{2d+9}{d+11}$

$\frac{12b+5}{3b\mathrm{-1}}$

$\frac{4y\mathrm{-1}}{y+4}$

$\frac{10x+4y}{xy}$

$\frac{9a-7}{a^2-2a-3}$

$\frac{2y^2-y+9}{y^2-y-2}$

$\frac{5z^2+z+5}{z^2-z-2}$

$\frac{x+2xy+y}{x+xy+y+1}$

$\frac{2b+7a}{a{b}^2}$

$\frac{18+ab}{4b}$

$a-b$

$\frac{3c^2+3c-2}{2c^2+5c+2}$

$\frac{15x+7}{x\mathrm{-1}}$

$\frac{x+9}{x\mathrm{-9}}$

$\frac{1}{y+2}$

$4$

$\mathrm{-5}$

53

$y=24$

$32m$

whole

irrational

$16$

$3a^6$

$\frac{x^3}{32y^3}$

$a$

$1.634\times {10}^7$

14

$5\sqrt{3}$

$\frac{4\sqrt{2}}{5}$

$\frac{7\sqrt{2}}{50}$

$10\sqrt{3}$

$\mathrm{-3}$