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Use the Definition of a Negative Exponent
In the following exercises, simplify.
ⓐ
${4}^{\mathrm{-2}}$
ⓑ
${10}^{\mathrm{-3}}$
ⓐ
${3}^{\mathrm{-4}}$
ⓑ
${10}^{\mathrm{-2}}$
ⓐ $\frac{1}{81}$ ⓑ $\frac{1}{100}$
ⓐ
${5}^{-3}$
ⓑ
${10}^{\mathrm{-5}}$
ⓐ
${2}^{\mathrm{-8}}$
ⓑ
${10}^{\mathrm{-2}}$
ⓐ $\frac{1}{256}$ ⓑ $\frac{1}{100}$
ⓐ
$\frac{1}{{c}^{\mathrm{-5}}}$
ⓑ
$\frac{1}{{3}^{\mathrm{-2}}}$
ⓐ
$\frac{1}{{c}^{\mathrm{-5}}}$
ⓑ
$\frac{1}{{5}^{\mathrm{-2}}}$
ⓐ ${c}^{5}$ ⓑ 25
ⓐ
$\frac{1}{{q}^{\mathrm{-10}}}$
ⓑ
$\frac{1}{{10}^{\mathrm{-3}}}$
ⓐ
$\frac{1}{{t}^{\mathrm{-9}}}$
ⓑ
$\frac{1}{{10}^{\mathrm{-4}}}$
ⓐ ${t}^{9}$ ⓑ 10000
ⓐ
${\left(\frac{5}{8}\right)}^{\mathrm{-2}}$
ⓑ
${\left(-\frac{3m}{n}\right)}^{\mathrm{-2}}$
ⓐ
${\left(\frac{3}{10}\right)}^{\mathrm{-2}}$
ⓑ
${\left(-\frac{2}{cd}\right)}^{\mathrm{-3}}$
ⓐ $\frac{100}{9}$ ⓑ $-\frac{{c}^{3}{d}^{3}}{8}$
ⓐ
${\left(\frac{4}{9}\right)}^{\mathrm{-3}}$
ⓑ
${\left(-\frac{{u}^{2}}{2v}\right)}^{\mathrm{-5}}$
ⓐ
${\left(\frac{7}{2}\right)}^{\mathrm{-3}}$
ⓑ
${\left(-\frac{3}{x{y}^{2}}\right)}^{\mathrm{-3}}$
ⓐ $\frac{8}{343}$ ⓑ $-\frac{{x}^{3}{y}^{6}}{27}$
ⓐ
${\left(-5\right)}^{-2}$
ⓑ
$\text{\u2212}{5}^{\mathrm{-2}}$
ⓒ
${\left(-\frac{1}{5}\right)}^{\mathrm{-2}}$
ⓓ
$\text{\u2212}{\left(\frac{1}{5}\right)}^{\mathrm{-2}}$
ⓐ
${\left(\mathrm{-7}\right)}^{\mathrm{-2}}$
ⓑ
$-{7}^{-2}$
ⓒ
${\left(-\frac{1}{7}\right)}^{\mathrm{-2}}$
ⓓ
$\text{\u2212}{\left(\frac{1}{7}\right)}^{\mathrm{-2}}$
ⓐ $\frac{1}{49}$ ⓑ $-\frac{1}{49}$ ⓒ 49 ⓓ $\mathrm{-49}$
ⓐ
$\text{\u2212}{3}^{\mathrm{-3}}$
ⓑ
${\left(-\frac{1}{3}\right)}^{\mathrm{-3}}$
ⓒ
$\text{\u2212}{\left(\frac{1}{3}\right)}^{\mathrm{-3}}$
ⓓ
${\left(\mathrm{-3}\right)}^{\mathrm{-3}}$
ⓐ
$\text{\u2212}{5}^{\mathrm{-3}}$
ⓑ
${\left(-\frac{1}{5}\right)}^{\mathrm{-3}}$
ⓒ
$\text{\u2212}{\left(\frac{1}{5}\right)}^{\mathrm{-3}}$
ⓓ
${\left(\mathrm{-5}\right)}^{\mathrm{-3}}$
ⓐ $-\frac{1}{125}$ ⓑ $\mathrm{-125}$ ⓒ $\mathrm{-125}$ ⓓ $-\frac{1}{125}$
ⓐ
$3\xb7{5}^{\mathrm{-1}}$
ⓑ
${\left(3\xb75\right)}^{\mathrm{-1}}$
ⓐ
$2\xb7{5}^{\mathrm{-1}}$
ⓑ
${\left(2\xb75\right)}^{\mathrm{-1}}$
ⓐ $\frac{2}{5}$ ⓑ $\frac{1}{10}$
ⓐ
$4\xb7{5}^{\mathrm{-2}}$
ⓑ
${\left(4\xb75\right)}^{\mathrm{-2}}$
ⓐ
$3\xb7{4}^{\mathrm{-2}}$
ⓑ
${\left(3\xb74\right)}^{\mathrm{-2}}$
ⓐ $\frac{3}{16}$ ⓑ $\frac{1}{144}$
ⓐ
${m}^{\mathrm{-4}}$
ⓑ
${\left({x}^{3}\right)}^{\mathrm{-4}}$
ⓐ
${b}^{\mathrm{-5}}$
ⓑ
${\left({k}^{2}\right)}^{\mathrm{-5}}$
ⓐ $\frac{1}{{b}^{5}}$ ⓑ $\frac{1}{{k}^{10}}$
ⓐ
${p}^{\mathrm{-10}}$
ⓑ
${\left({q}^{6}\right)}^{\mathrm{-8}}$
ⓐ
${s}^{\mathrm{-8}}$
ⓑ
${\left({a}^{9}\right)}^{\mathrm{-10}}$
ⓐ $\frac{1}{{s}^{8}}$ ⓑ $\frac{1}{{a}^{90}}$
ⓐ
$7{n}^{\mathrm{-1}}$
ⓑ
${(7n)}^{\mathrm{-1}}$
ⓒ
${(\mathrm{-7}n)}^{\mathrm{-1}}$
ⓐ
$6{r}^{\mathrm{-1}}$
ⓑ
${(6r)}^{\mathrm{-1}}$
ⓒ
${(\mathrm{-6}r)}^{\mathrm{-1}}$
ⓐ $\frac{6}{r}$ ⓑ $\frac{1}{6r}$ ⓒ $-\frac{1}{6r}$
ⓐ
${\left(3p\right)}^{\mathrm{-2}}$
ⓑ
$3{p}^{\mathrm{-2}}$
ⓒ
$\mathrm{-3}{p}^{\mathrm{-2}}$
ⓐ
${\left(2q\right)}^{\mathrm{-4}}$
ⓑ
$2{q}^{\mathrm{-4}}$
ⓒ
$\mathrm{-2}{q}^{\mathrm{-4}}$
ⓐ $\frac{1}{16{q}^{4}}$ ⓑ $\frac{2}{{q}^{4}}$ ⓒ $-\frac{2}{{q}^{4}}$
Simplify Expressions with Integer Exponents
In the following exercises, simplify.
ⓐ
${b}^{4}{b}^{\mathrm{-8}}$
ⓑ
${r}^{\mathrm{-2}}{r}^{5}$
ⓒ
${x}^{\mathrm{-7}}{x}^{\mathrm{-3}}$
ⓐ
${s}^{3}\xb7{s}^{\mathrm{-7}}$
ⓑ
${q}^{\mathrm{-8}}\xb7{q}^{3}$
ⓒ
${y}^{\mathrm{-2}}\xb7{y}^{\mathrm{-5}}$
ⓐ $\frac{1}{{s}^{4}}$ ⓑ $\frac{1}{{q}^{5}}$ ⓒ $\frac{1}{{y}^{7}}$
ⓐ
${a}^{3}\xb7{a}^{\mathrm{-3}}$
ⓑ
$a\xb7{a}^{3}$
ⓒ
$a\xb7{a}^{\mathrm{-3}}$
ⓐ
${y}^{5}\xb7{y}^{\mathrm{-5}}$
ⓑ
$y\xb7{y}^{5}$
ⓒ
$y\xb7{y}^{\mathrm{-5}}$
ⓐ 1 ⓑ ${y}^{6}$ ⓒ $\frac{1}{{y}^{4}}$
${p}^{5}\xb7{p}^{\mathrm{-2}}\xb7{p}^{\mathrm{-4}}$
${x}^{4}\xb7{x}^{\mathrm{-2}}\xb7{x}^{\mathrm{-3}}$
$\frac{1}{x}$
$\left({w}^{4}{x}^{\mathrm{-5}}\right)\left({w}^{\mathrm{-2}}{x}^{\mathrm{-4}}\right)$
$\left({m}^{3}{n}^{\mathrm{-3}}\right)\left({m}^{\mathrm{-5}}{n}^{\mathrm{-1}}\right)$
$\frac{1}{{m}^{2}{n}^{4}}$
$\left(u{v}^{\mathrm{-2}}\right)\left({u}^{\mathrm{-5}}{v}^{\mathrm{-3}}\right)$
$\left(p{q}^{\mathrm{-4}}\right)\left({p}^{\mathrm{-6}}{q}^{\mathrm{-3}}\right)$
$\frac{1}{{p}^{5}{q}^{7}}$
$\left(\mathrm{-6}{c}^{\mathrm{-3}}{d}^{9}\right)\left(2{c}^{4}{d}^{\mathrm{-5}}\right)$
$\left(\mathrm{-2}{j}^{\mathrm{-5}}{k}^{8}\right)\left(7{j}^{2}{k}^{\mathrm{-3}}\right)$
$-\frac{14{k}^{5}}{{j}^{3}}$
$\left(\mathrm{-4}{r}^{\mathrm{-2}}{s}^{\mathrm{-8}}\right)\left(9{r}^{4}{s}^{3}\right)$
$\left(\mathrm{-5}{m}^{4}{n}^{6}\right)\left(8{m}^{\mathrm{-5}}{n}^{\mathrm{-3}}\right)$
$-\frac{40{n}^{3}}{m}$
${\left(5{x}^{2}\right)}^{\mathrm{-2}}$
${\left(4{y}^{3}\right)}^{\mathrm{-3}}$
$\frac{1}{64{y}^{9}}$
${\left(3{z}^{\mathrm{-3}}\right)}^{2}$
${\left(2{p}^{\mathrm{-5}}\right)}^{2}$
$\frac{4}{{p}^{10}}$
$\frac{{t}^{9}}{{t}^{\mathrm{-3}}}$
$\frac{{x}^{\mathrm{-7}}}{{x}^{\mathrm{-3}}}$
Convert from Decimal Notation to Scientific Notation
In the following exercises, write each number in scientific notation.
340,000
$3.4\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}$
8,750,000
1,290,000
$1.29\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{6}$
0.041
$4.1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-2}}$
0.00000871
0.00000103
$1.03\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-6}}$
Convert Scientific Notation to Decimal Form
In the following exercises, convert each number to decimal form.
$5.2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{2}$
$8.3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{2}$
830
$7.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{6}$
$1.6\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{10}$
16,000,000,000
$2.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-2}}$
$3.8\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-2}}$
0.038
$4.13\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-5}}$
$1.93\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-5}}$
0.0000193
Multiply and Divide Using Scientific Notation
In the following exercises, multiply. Write your answer in decimal form.
$\left(3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-5}}\right)\left(3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{9}\right)$
$\left(2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{2}\right)\left(1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}}\right)$
0.02
$\left(7.1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-2}}\right)\left(2.4\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}}\right)$
$\left(3.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}}\right)\left(1.6\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-2}}\right)$
$5.6\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-6}}$
In the following exercises, divide. Write your answer in decimal form.
$\frac{7\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-3}}}{1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-7}}}$
$\frac{5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-2}}}{1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-10}}}$
500,000,000
$\frac{6\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{4}}{3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-2}}}$
$\frac{8\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{6}}{4\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-1}}}$
20,000,000
The population of the United States on July 4, 2010 was almost 310,000,000. Write the number in scientific notation.
The population of the world on July 4, 2010 was more than 6,850,000,000. Write the number in scientific notation
$6.85\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{9}$ .
The average width of a human hair is 0.0018 centimeters. Write the number in scientific notation.
The probability of winning the 2010 Megamillions lottery was about 0.0000000057. Write the number in scientific notation.
$5.7\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-10}}$
In 2010, the number of Facebook users each day who changed their status to ‘engaged’ was $2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{4}$ . Convert this number to decimal form.
At the start of 2012, the US federal budget had a deficit of more than $\text{\$}1.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{13}$ . Convert this number to decimal form.
15,000,000,000,000
The concentration of carbon dioxide in the atmosphere is $3.9\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}}$ . Convert this number to decimal form.
The width of a proton is $1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-5}}$ of the width of an atom. Convert this number to decimal form.
0.00001
Health care costs The Centers for Medicare and Medicaid projects that consumers will spend more than $4 trillion on health care by 2017.
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