1.2 Exponents and Scientific Notation

Write each of the following products with a single base. Do not simplify further.

1. ⓐ $t^{5} \cdot t^{3}$
2. ⓑ $\left(−3\right)^{5} \cdot \left(−3\right)$
3. ⓒ $x^{2} \cdot x^{5} \cdot x^{3}$

Write each of the following products with a single base. Do not simplify further.

1. ⓐ $k^{6} \cdot k^{9}$
2. ⓑ $\left(\frac{2}{y}\right)^{4} \cdot \left(\frac{2}{y}\right)$
3. ⓒ $t^{3} \cdot t^{6} \cdot t^{5}$

Write each of the following products with a single base. Do not simplify further.

1. ⓐ $\frac{\left(−2\right)^{14}}{\left(−2\right)^{9}}$
2. ⓑ $\frac{t^{23}}{t^{15}}$
3. ⓒ $\frac{\left(z \sqrt{2}\right)^{5}}{z \sqrt{2}}$

Write each of the following products with a single base. Do not simplify further.

1. ⓐ $\frac{s^{75}}{s^{68}}$
2. ⓑ $\frac{\left(−3\right)^{6}}{−3}$
3. ⓒ $\frac{\left(e f^{2}\right)^{5}}{\left(e f^{2}\right)^{3}}$

Write each of the following products with a single base. Do not simplify further.

1. ⓐ $\left(x^{2}\right)^{7}$
2. ⓑ $\left(\left(2 t\right)^{5}\right)^{3}$
3. ⓒ $\left(\left(−3\right)^{5}\right)^{11}$

Write each of the following products with a single base. Do not simplify further.

1. ⓐ $\left(\left(3 y\right)^{8}\right)^{3}$
2. ⓑ $\left(t^{5}\right)^{7}$
3. ⓒ $\left(\left(- g\right)^{4}\right)^{4}$

Simplify each expression using the zero exponent rule of exponents.

1. ⓐ $\frac{c^{3}}{c^{3}}$
2. ⓑ $\frac{−3 x^{5}}{x^{5}}$
3. ⓒ $\frac{\left(j^{2} k\right)^{4}}{\left(j^{2} k\right) \cdot \left(j^{2} k\right)^{3}}$
4. ⓓ $\frac{5 \left(r s^{2}\right)^{2}}{\left(r s^{2}\right)^{2}}$

Simplify each expression using the zero exponent rule of exponents.

1. ⓐ $\frac{t^{7}}{t^{7}}$
2. ⓑ $\frac{\left(d e^{2}\right)^{11}}{2 \left(d e^{2}\right)^{11}}$
3. ⓒ $\frac{w^{4} \cdot w^{2}}{w^{6}}$
4. ⓓ $\frac{t^{3} \cdot t^{4}}{t^{2} \cdot t^{5}}$

Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.

1. ⓐ $\frac{\theta^{3}}{\theta^{10}}$
2. ⓑ $\frac{z^{2} \cdot z}{z^{4}}$
3. ⓒ $\frac{\left(−5 t^{3}\right)^{4}}{\left(−5 t^{3}\right)^{8}}$

Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.

1. ⓐ $\frac{\left(−3 t\right)^{2}}{\left(−3 t\right)^{8}}$
2. ⓑ $\frac{f^{47}}{f^{49} \cdot f}$
3. ⓒ $\frac{2 k^{4}}{5 k^{7}}$

Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.

1. ⓐ $b^{2} \cdot b^{−8}$
2. ⓑ $\left(- x\right)^{5} \cdot \left(- x\right)^{−5}$
3. ⓒ $\frac{−7 z}{\left(−7 z\right)^{5}}$

Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.

1. ⓐ $t^{−11} \cdot t^{6}$
2. ⓑ $\frac{25^{12}}{25^{13}}$

Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.

1. ⓐ $\left(a b^{2}\right)^{3}$
2. ⓑ $\left(2 t\right)^{15}$
3. ⓒ $\left(−2 w^{3}\right)^{3}$
4. ⓓ $\frac{1}{\left(−7 z\right)^{4}}$
5. ⓔ $\left(e^{−2} f^{2}\right)^{7}$

Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.

1. ⓐ $\left(g^{2} h^{3}\right)^{5}$
2. ⓑ $\left(5 t\right)^{3}$
3. ⓒ $\left(−3 y^{5}\right)^{3}$
4. ⓓ $\frac{1}{\left(a^{6} b^{7}\right)^{3}}$
5. ⓔ $\left(r^{3} s^{−2}\right)^{4}$

Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.

1. ⓐ $\left(\frac{4}{z^{11}}\right)^{3}$
2. ⓑ $\left(\frac{p}{q^{3}}\right)^{6}$
3. ⓒ $\left(\frac{−1}{t^{2}}\right)^{27}$
4. ⓓ $\left(j^{3} k^{−2}\right)^{4}$
5. ⓔ $\left(m^{−2} n^{−2}\right)^{3}$

Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.

1. ⓐ $\left(\frac{b^{5}}{c}\right)^{3}$
2. ⓑ $\left(\frac{5}{u^{8}}\right)^{4}$
3. ⓒ $\left(\frac{−1}{w^{3}}\right)^{35}$
4. ⓓ $\left(p^{−4} q^{3}\right)^{8}$
5. ⓔ $\left(c^{−5} d^{−3}\right)^{4}$

Simplify each expression and write the answer with positive exponents only.

1. ⓐ $\left(6 m^{2} n^{−1}\right)^{3}$
2. ⓑ $17^{5} \cdot 17^{−4} \cdot 17^{−3}$
3. ⓒ $\left(\frac{u^{−1} v}{v^{−1}}\right)^{2}$
4. ⓓ $\left(−2 a^{3} b^{−1}\right) \left(5 a^{−2} b^{2}\right)$
5. ⓔ $\left(x^{2} \sqrt{2}\right)^{4} \left(x^{2} \sqrt{2}\right)^{−4}$
6. ⓕ $\frac{\left(3 w^{2}\right)^{5}}{\left(6 w^{−2}\right)^{2}}$

Simplify each expression and write the answer with positive exponents only.

1. ⓐ $\left(2 u v^{- 2}\right)^{−3}$
2. ⓑ $x^{8} \cdot x^{−12} \cdot x$
3. ⓒ $\left(\frac{e^{2} f^{- 3}}{f^{−1}}\right)^{2}$
4. ⓓ $\left(9 r^{−5} s^{3}\right) \left(3 r^{6} s^{−4}\right)$
5. ⓔ $\left(\frac{4}{9} t w^{−2}\right)^{−3} \left(\frac{4}{9} t w^{−2}\right)^{3}$
6. ⓕ $\frac{\left(2 h^{2} k\right)^{4}}{\left(7 h^{−1} k^{2}\right)^{2}}$

Write each number in scientific notation.

ⓐDistance to Andromeda Galaxy from Earth: 24,000,000,000,000,000,000,000 m

ⓑDiameter of Andromeda Galaxy: 1,300,000,000,000,000,000,000 m

ⓒNumber of stars in Andromeda Galaxy: 1,000,000,000,000

ⓓDiameter of electron: 0.00000000000094 m

ⓔProbability of being struck by lightning in any single year: 0.00000143

Write each number in scientific notation.

1. ⓐU.S. national debt per taxpayer (April 2014): $152,000
2. ⓑWorld population (April 2014): 7,158,000,000
3. ⓒWorld gross national income (April 2014): $85,500,000,000,000
4. ⓓTime for light to travel 1 m: 0.00000000334 s
5. ⓔProbability of winning lottery (match 6 of 49 possible numbers): 0.0000000715

Convert each number in scientific notation to standard notation.

1. ⓐ $3.547 \times 10^{14}$
2. ⓑ $−2 \times 10^{6}$
3. ⓒ $7.91 \times 10^{−7}$
4. ⓓ $−8.05 \times 10^{−12}$

Convert each number in scientific notation to standard notation.

1. ⓐ $7.03 \times 10^{5}$
2. ⓑ $−8.16 \times 10^{11}$
3. ⓒ $−3.9 \times 10^{−13}$
4. ⓓ $8 \times 10^{−6}$

Perform the operations and write the answer in scientific notation.

1. ⓐ $\left(8.14 \times 10^{−7}\right) \left(6.5 \times 10^{10}\right)$
2. ⓑ $\left(4 \times 10^{5}\right) \div \left(−1.52 \times 10^{9}\right)$
3. ⓒ $\left(2.7 \times 10^{5}\right) \left(6.04 \times 10^{13}\right)$
4. ⓓ $\left(1.2 \times 10^{8}\right) \div \left(9.6 \times 10^{5}\right)$
5. ⓔ $\left(3.33 \times 10^{4}\right) \left(−1.05 \times 10^{7}\right) \left(5.62 \times 10^{5}\right)$

Perform the operations and write the answer in scientific notation.

1. ⓐ $\left(−7.5 \times 10^{8}\right) \left(1.13 \times 10^{−2}\right)$
2. ⓑ $\left(1.24 \times 10^{11}\right) \div \left(1.55 \times 10^{18}\right)$
3. ⓒ $\left(3.72 \times 10^{9}\right) \left(8 \times 10^{3}\right)$
4. ⓓ $\left(9.933 \times 10^{23}\right) \div \left(- 2.31 \times 10^{17}\right)$
5. ⓔ $\left(−6.04 \times 10^{9}\right) \left(7.3 \times 10^{2}\right) \left(−2.81 \times 10^{2}\right)$

In April 2014, the population of the United States was about 308,000,000 people. The national debt was about $17,547,000,000,000. Write each number in scientific notation, rounding figures to two decimal places, and find the amount of the debt per U.S. citizen. Write the answer in both scientific and standard notations.

An average human body contains around 30,000,000,000,000 red blood cells. Each cell measures approximately 0.000008 m long. Write each number in scientific notation and find the total length if the cells were laid end-to-end. Write the answer in both scientific and standard notations.

Is $2^{3}$ the same as $3^{2} ?$ Explain.

What is the purpose of scientific notation?

Simplify the given expression. Write answers with positive exponents.

$9^{2}$

Simplify the given expression. Write answers with positive exponents.

$3^{2} \times 3^{3}$

Simplify the given expression. Write answers with positive exponents.

$\left(2^{2}\right)^{−2}$

Simplify the given expression. Write answers with positive exponents.

$11^{3} \div 11^{4}$

Simplify the given expression. Write answers with positive exponents.

$\left(8^{0}\right)^{2}$

Write each expression with a single base. Do not simplify further. Write answers with positive exponents.

$4^{2} \times 4^{3} \div 4^{−4}$

Write each expression with a single base. Do not simplify further. Write answers with positive exponents.

$\left(12^{3} \times 12\right)^{10}$

Write each expression with a single base. Do not simplify further. Write answers with positive exponents.

$7^{−6} \times 7^{−3}$

Express the decimal in scientific notation.

0.0000314

Convert each number in scientific notation to standard notation.

$1.6 \times 10^{10}$

Simplify the given expression. Write answers with positive exponents.

$\frac{a^{3} a^{2}}{a}$

Simplify the given expression. Write answers with positive exponents.

$\left(b^{3} c^{4}\right)^{2}$

Simplify the given expression. Write answers with positive exponents.

$a b^{2} \div d^{−3}$

Simplify the given expression. Write answers with positive exponents.

$\frac{m^{4}}{n^{0}}$

Simplify the given expression. Write answers with positive exponents.

$\frac{p^{−4} q^{2}}{p^{2} q^{−3}}$

Simplify the given expression. Write answers with positive exponents.

$\left(y^{7}\right)^{3} \div x^{14}$

Simplify the given expression. Write answers with positive exponents.

$\left(25 m\right) \div \left(5^{0} m\right)$

Simplify the given expression. Write answers with positive exponents.

$\frac{2^{3}}{\left(3 a\right)^{−2}}$

Simplify the given expression. Write answers with positive exponents.

$\left(b^{−3} c\right)^{3}$

Simplify the given expression. Write answers with positive exponents.

$\left(9 z^{3}\right)^{−2} y$

A dime is the thinnest coin in U.S. currency. A dime’s thickness measures $1.35 \times 10^{−3}$ m. Rewrite the number in standard notation.

A terabyte is made of approximately 1,099,500,000,000 bytes. Rewrite in scientific notation.

One picometer is approximately $3.397 \times 10^{−11}$ in. Rewrite this length using standard notation.

Use a graphing calculator to simplify. Round the answers to the nearest hundredth.

$\left(\frac{12^{3} m^{33}}{4^{−3}}\right)^{2}$

Simplify the given expression. Write answers with positive exponents.

$\left(\frac{3^{2}}{a^{3}}\right)^{−2} \left(\frac{a^{4}}{2^{2}}\right)^{2}$

Simplify the given expression. Write answers with positive exponents.

$\frac{m^{2} n^{3}}{a^{2} c^{−3}} \cdot \frac{a^{−7} n^{−2}}{m^{2} c^{4}}$

Simplify the given expression. Write answers with positive exponents.

$\left(\frac{\left(a b^{2} c\right)^{−3}}{b^{−3}}\right)^{2}$

Simplify the given expression. Write answers with positive exponents.

Planck’s constant is an important unit of measure in quantum physics. It describes the relationship between energy and frequency. The constant is written as $6.62606957 \times 10^{−34} .$ Write Planck’s constant in standard notation.