6.7 Integer Exponents and Scientific Notation

Before you get started, take this readiness quiz.

What is the place value of the $6$ in the number $64,891$?
If you missed this problem, review the reference.

Name the decimal: $0.0012 .$
If you missed this problem, review the reference.

Subtract: $5 - \left(−3\right) .$
If you missed this problem, review the reference.

Simplify: ⓐ $4^{−2}$ ⓑ $10^{−3} .$

Simplify: ⓐ $2^{−3}$ ⓑ $10^{−7} .$

Simplify: ⓐ $3^{−2}$ ⓑ $10^{−4} .$

Simplify: ⓐ $\frac{1}{y^{−4}}$ ⓑ $\frac{1}{3^{−2}} .$

Simplify: ⓐ $\frac{1}{p^{−8}}$ ⓑ $\frac{1}{4^{−3}} .$

Simplify: ⓐ $\frac{1}{q^{−7}}$ ⓑ $\frac{1}{2^{−4}} .$

Simplify: ⓐ $\left(\frac{5}{7}\right)^{−2}$ ⓑ $\left(- \frac{2 x}{y}\right)^{−3} .$

Simplify: ⓐ $\left(\frac{2}{3}\right)^{−4}$ ⓑ $\left(- \frac{6 m}{n}\right)^{−2} .$

Simplify: ⓐ $\left(\frac{3}{5}\right)^{−3}$ ⓑ $\left(- \frac{a}{2 b}\right)^{−4} .$

Simplify: ⓐ $\left(−3\right)^{−2}$ ⓑ $− 3^{−2}$ ⓒ $\left(- \frac{1}{3}\right)^{−2}$ ⓓ $− \left(\frac{1}{3}\right)^{−2} .$

Simplify: ⓐ $\left(−5\right)^{−2}$ ⓑ $− 5^{−2}$ ⓒ $\left(- \frac{1}{5}\right)^{−2}$ ⓓ $− \left(\frac{1}{5}\right)^{−2} .$

Simplify: ⓐ $\left(−7\right)^{−2}$ ⓑ $− 7^{−2}$, ⓒ $\left(- \frac{1}{7}\right)^{−2}$ ⓓ $− \left(\frac{1}{7}\right)^{−2} .$

Simplify: ⓐ $4 \cdot 2^{−1}$ ⓑ $\left(4 \cdot 2\right)^{−1} .$

Simplify: ⓐ $6 \cdot 3^{−1}$ ⓑ $\left(6 \cdot 3\right)^{−1} .$

Simplify: ⓐ $8 \cdot 2^{−2}$ ⓑ $\left(8 \cdot 2\right)^{−2} .$

Simplify: ⓐ $x^{−6}$ ⓑ $\left(u^{4}\right)^{−3} .$

Simplify: ⓐ $y^{−7}$ ⓑ $\left(z^{3}\right)^{−5} .$

Simplify: ⓐ $p^{−9}$ ⓑ $\left(q^{4}\right)^{−6} .$

Simplify: ⓐ $5 y^{−1}$ ⓑ $\left(5 y\right)^{−1}$ ⓒ $\left(−5 y\right)^{−1} .$

Simplify: ⓐ $8 p^{−1}$ ⓑ $\left(8 p\right)^{−1}$ ⓒ $\left(−8 p\right)^{−1} .$

Simplify: ⓐ $11 q^{−1}$ ⓑ $\left(11 q\right)^{−1}$$− \left(11 q\right)^{−1}$ ⓒ $- \frac{1}{11 q}$

Simplify: ⓐ $x^{−4} \cdot x^{6}$ ⓑ $y^{−6} \cdot y^{4}$ ⓒ $z^{−5} \cdot z^{−3} .$

Simplify: ⓐ $x^{−3} \cdot x^{7}$ ⓑ $y^{−7} \cdot y^{2}$ ⓒ $z^{−4} \cdot z^{−5} .$

Simplify: ⓐ $a^{−1} \cdot a^{6}$ ⓑ $b^{−8} \cdot b^{4}$ ⓒ $c^{−8} \cdot c^{−7} .$

Simplify: $\left(m^{4} n^{−3}\right) \left(m^{−5} n^{−2}\right) .$

Simplify: $\left(p^{6} q^{−2}\right) \left(p^{−9} q^{−1}\right) .$

Simplify: $\left(r^{5} s^{−3}\right) \left(r^{−7} s^{−5}\right) .$

Simplify: $\left(−6 c^{−6} d^{4}\right) \left(−5 c^{−2} d^{−1}\right) .$

Simplify: $\left(6 k^{3}\right)^{−2} .$

Simplify: $\left(−4 x^{4}\right)^{−2} .$

Simplify: $\left(2 b^{3}\right)^{−4} .$

Simplify: $\left(5 x^{−3}\right)^{2} .$

Simplify: $\left(8 a^{−4}\right)^{2} .$

Simplify: $\left(2 c^{−4}\right)^{3} .$

Simplify: $\frac{r^{5}}{r^{−4}} .$

Simplify: $\frac{x^{8}}{x^{−3}} .$

Simplify: $\frac{y^{8}}{y^{−6}} .$

Write in scientific notation: $96,000 .$

Write in scientific notation: $48,300 .$

Write in scientific notation: $0.0078 .$

Write in scientific notation: $0.0129 .$

Convert to decimal form: $1.3 \times 10^{3} .$

Convert to decimal form: $9.25 \times 10^{4} .$

Convert to decimal form: $1.2 \times 10^{−4} .$

Convert to decimal form: $7.5 \times 10^{−2} .$

Multiply. Write answers in decimal form: $\left(4 \times 10^{5}\right) \left(2 \times 10^{−7}\right) .$

Multiply $\left(3 \times 10^{6}\right) \left(2 \times 10^{−8}\right)$. Write answers in decimal form.

Multiply $\left(3 \times 10^{−2}\right) \left(3 \times 10^{−1}\right)$. Write answers in decimal form.

Divide. Write answers in decimal form: $\frac{9 \times 10^{3}}{3 \times 10^{−2}} .$

Divide $\frac{8 \times 10^{4}}{2 \times 10^{−1}}$. Write answers in decimal form.

Divide $\frac{8 \times 10^{2}}{4 \times 10^{−2}}$. Write answers in decimal form.

Simplify.

ⓐ $3^{−4}$ ⓑ $10^{−2}$

Simplify.

ⓐ $2^{−8}$ ⓑ $10^{−2}$

Simplify.

ⓐ $\frac{1}{c^{−5}}$ ⓑ $\frac{1}{5^{−2}}$

Simplify.

ⓐ $\frac{1}{t^{−9}}$ ⓑ $\frac{1}{10^{−4}}$

Simplify.

ⓐ $\left(\frac{3}{10}\right)^{−2}$ ⓑ $\left(- \frac{2}{c d}\right)^{−3}$

Simplify.

ⓐ $\left(\frac{7}{2}\right)^{−3}$ ⓑ $\left(- \frac{3}{x y^{2}}\right)^{−3}$

Simplify.

ⓐ $\left(−7\right)^{−2}$ ⓑ $- 7^{- 2}$ ⓒ $\left(- \frac{1}{7}\right)^{−2}$ ⓓ $− \left(\frac{1}{7}\right)^{−2}$

Simplify.

ⓐ $− 5^{−3}$ ⓑ $\left(- \frac{1}{5}\right)^{−3}$ ⓒ $− \left(\frac{1}{5}\right)^{−3}$ ⓓ $\left(−5\right)^{−3}$

Simplify.

ⓐ $2 \cdot 5^{−1}$ ⓑ $\left(2 \cdot 5\right)^{−1}$

Simplify.

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

Simplify.

ⓐ $b^{−5}$ ⓑ $\left(k^{2}\right)^{−5}$

Simplify.

ⓐ $s^{−8}$ ⓑ $\left(a^{9}\right)^{−10}$

Simplify.

ⓐ $6 r^{−1}$ ⓑ $\left(6 r\right)^{−1}$ ⓒ $\left(−6 r\right)^{−1}$

Simplify.

ⓐ $\left(2 q\right)^{−4}$ ⓑ $2 q^{−4}$ ⓒ $−2 q^{−4}$

Simplify.

ⓐ $s^{3} \cdot s^{−7}$ ⓑ $q^{−8} \cdot q^{3}$ ⓒ $y^{−2} \cdot y^{−5}$

Simplify.

ⓐ $y^{5} \cdot y^{−5}$ ⓑ $y \cdot y^{5}$ ⓒ $y \cdot y^{−5}$

Simplify.

$x^{4} \cdot x^{−2} \cdot x^{−3}$

Simplify.

$\left(m^{3} n^{−3}\right) \left(m^{−5} n^{−1}\right)$

Simplify.

$\left(p q^{−4}\right) \left(p^{−6} q^{−3}\right)$

Simplify.

$\left(−2 j^{−5} k^{8}\right) \left(7 j^{2} k^{−3}\right)$

Simplify.

$\left(−5 m^{4} n^{6}\right) \left(8 m^{−5} n^{−3}\right)$

Simplify.

$\left(4 y^{3}\right)^{−3}$

Simplify.

$\left(2 p^{−5}\right)^{2}$

Simplify.

$\frac{n^{5}}{n^{−2}}$

Simplify.

$\frac{y^{−5}}{y^{−10}}$

Write each number in scientific notation.

340,000

Write each number in scientific notation.

1,290,000

Write each number in scientific notation.

0.041

Write each number in scientific notation.

0.00000103

Convert each number to decimal form.

$8.3 \times 10^{2}$

Convert each number to decimal form.

$1.6 \times 10^{10}$

Convert each number to decimal form.

$3.8 \times 10^{−2}$

Convert each number to decimal form.

$1.93 \times 10^{−5}$

Multiply. Write your answer in decimal form.

$\left(2 \times 10^{2}\right) \left(1 \times 10^{−4}\right)$

Multiply. Write your answer in decimal form.

$\left(3.5 \times 10^{−4}\right) \left(1.6 \times 10^{−2}\right)$

Divide. Write your answer in decimal form.

$\frac{5 \times 10^{−2}}{1 \times 10^{−10}}$

Divide. Write your answer in decimal form.

$\frac{8 \times 10^{6}}{4 \times 10^{−1}}$

The population of the world on July 4, 2010 was more than 6,850,000,000. Write the number in scientific notation

The probability of winning the 2010 Megamillions lottery was about 0.0000000057. Write the number in scientific notation.

At the start of 2012, the US federal budget had a deficit of more than $\$ 1.5 \times 10^{13}$. Convert this number to decimal form.

The width of a proton is $1 \times 10^{−5}$ of the width of an atom. Convert this number to decimal form.

Coin production In 1942, the U.S. Mint produced 154,500,000 nickels. Write 154,500,000 in scientific notation.

Debt At the end of fiscal year 2015 the gross United States federal government debt was estimated to be approximately $18,600,000,000,000 ($18.6 trillion), according to the Federal Budget. The population of the United States was approximately 300,000,000 people at the end of fiscal year 2015.

1. ⓐ Write the debt in scientific notation.
2. ⓑ Write the population in scientific notation.
3. ⓒ Find the amount…

When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?

Determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

ⓐ $a^{2} - b^{2}$
ⓑ $24 d^{3}$
ⓒ $x^{2} + 8 x - 10$
ⓓ $m^{2} n^{2} - 2 m n + 6$
ⓔ $7 y^{3} + y^{2} - 2 y - 4$

Determine the degree of each polynomial.

1. ⓐ $5 p^{3} - 8 p^{2} + 10 p - 4$
2. ⓑ $−20 q^{4}$
3. ⓒ $x^{2} + 6 x + 12$
4. ⓓ $23 r^{2} s^{2} - 4 r s + 5$
5. ⓔ 100

Add or subtract the monomials.

$−14 k + 19 k$

Add or subtract the monomials.

$−9 c - 18 c$

Add or subtract the monomials.

$3 m^{2} + 7 n^{2} - 3 m^{2}$

Add or subtract the monomials.

$\text{13a} + b$

Add or subtract the polynomials.

$\left(9 p^{2} - 5 p + 3\right) + \left(4 p^{2} - 4\right)$

Add or subtract the polynomials.

$\left(7 y^{2} - 8 y\right) - \left(y - 4\right)$

Add or subtract the polynomials.

Find the sum of $\left(a^{2} + 6 a + 9\right) \text{and} \left(5 a^{3} - 7\right)$

Evaluate each polynomial for the given value.

Evaluate $10 - 12 x$ when:

1. ⓐ $x = 3$
2. ⓑ $x = 0$
3. ⓒ $x = −1$

Evaluate each polynomial for the given value.

A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial $−4 p^{2} + 460 p .$ Find the revenue received when $p = 75$ dollars.

Simplify.

$17^{1}$

Simplify.

$\left(0.5\right)^{3}$

Simplify.

$− 2^{6}$

Simplify each expression.

$p^{15} \cdot p^{16}$

Simplify each expression.

$8 \cdot 8^{5}$

Simplify each expression.

$y^{c} \cdot y^{3}$

Simplify each expression.

$\left(5^{3}\right)^{2}$

Simplify each expression.

$\left(3^{r}\right)^{s}$

Simplify each expression.

$\left(−5 y\right)^{3}$

Simplify each expression.

$\left(10 x y z\right)^{3}$

Simplify each expression.

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

Simplify each expression.

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

Simplify each expression.

$\left(\frac{2}{5} m^{2} n\right)^{3}$

In the following exercises 8, multiply the monomials.

$\left(−9 n^{7}\right) \left(−16 n\right)$

In the following exercises 8, multiply the monomials.

$\left(\frac{5}{9} a b^{2}\right) \left(27 a b^{3}\right)$

Multiply.

$−4 \left(y + 13\right)$

Multiply.

$p \left(p + 3\right)$

Multiply.

$−6 u \left(2 u + 7\right)$

Multiply.

$3 q^{2} \left(q^{2} - 7 q + 6\right)$ 3

Multiply.

$\left(b - 4\right) \cdot 11$

Multiply the binomials using: ⓐ the Distributive Property, ⓑ the FOIL method, ⓒ the Vertical Method.

$\left(6 y - 7\right) \left(2 y - 5\right)$

Multiply the binomials. Use any method.

$\left(y - 4\right) \left(y - 8\right)$

Multiply the binomials. Use any method.

$\left(q + 16\right) \left(q - 3\right)$

Multiply the binomials. Use any method.

$\left(u^{2} + 6\right) \left(u^{2} - 5\right)$

Multiply the binomials. Use any method.

$\left(8 m n + 3\right) \left(2 m n - 1\right)$

Multiply using ⓐ the Distributive Property, ⓑ the Vertical Method.

$\left(3 x - 4\right) \left(6 x^{2} + x - 10\right)$

Multiply. Use either method.

$\left(7 m + 1\right) \left(m^{2} - 10 m - 3\right)$

Square each binomial using the Binomial Squares Pattern.

$\left(q - 15\right)^{2}$

Square each binomial using the Binomial Squares Pattern.

$\left(8 u + 1\right)^{2}$

Square each binomial using the Binomial Squares Pattern.

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

Multiply each pair of conjugates using the Product of Conjugates Pattern.

$\left(y + \frac{2}{5}\right) \left(y - \frac{2}{5}\right)$

Multiply each pair of conjugates using the Product of Conjugates Pattern.

$\left(6 - r\right) \left(6 + r\right)$

Multiply each pair of conjugates using the Product of Conjugates Pattern.

$\left(5 p^{4} - 4 q^{3}\right) \left(5 p^{4} + 4 q^{3}\right)$

Find each product.

$\left(6 a + 11\right) \left(6 a - 11\right)$

Find each product.

$\left(c^{4} + 9 d\right)^{2}$

Find each product.

$\left(a^{2} + 4 b\right) \left(4 a - b^{2}\right)$

Simplify.

$\frac{10^{25}}{10^{5}}$

Simplify.

$\frac{v^{12}}{v^{48}}$

Simplify.

$\frac{5}{5^{8}}$

Simplify.

$x^{0}$

Simplify.

$\left(− 12^{0}\right)$$\left(−12\right)^{0}$

Simplify.

$\left(25 x\right)^{0}$

Simplify.

$\left(19 n\right)^{0} - \left(25 m\right)^{0}$

Simplify.

$\left(\frac{m}{3}\right)^{4}$

Simplify.

$\left(\frac{x}{2 y}\right)^{6}$

Simplify.

$\frac{n^{10}}{\left(n^{5}\right)^{2}}$

Simplify.

$\left(\frac{r^{8}}{r^{3}}\right)^{4}$

Simplify.

$\left(\frac{3 x^{4}}{2 y^{2}}\right)^{5}$

Simplify.

$\frac{\left(3 n^{2}\right)^{4} \left(−5 n^{4}\right)^{3}}{\left(−2 n^{5}\right)^{2}}$

Divide the monomials.

$\frac{64 a^{5} b^{9}}{−16 a^{10} b^{3}}$

Divide the monomials.

$\frac{\left(8 p^{6} q^{2}\right) \left(9 p^{3} q^{5}\right)}{16 p^{8} q^{7}}$

Divide each polynomial by the monomial.

$\left(35 x^{2} - 75 x\right) \div 5 x$

Divide each polynomial by the monomial.

$\frac{550 p^{6} - 300 p^{4}}{10 p^{3}}$

Divide each polynomial by the monomial.

$\frac{96 a^{5} b^{2} - 48 a^{4} b^{3} - 56 a^{2} b^{4}}{8 a b^{2}}$

Divide each polynomial by the monomial.

$\frac{105 y^{5} + 50 y^{3} - 5 y}{5 y^{3}}$

Divide each polynomial by the binomial.

$\left(v^{2} - 16 v + 64\right) \div \left(v - 8\right)$

Divide each polynomial by the binomial.

$\left(n^{2} - 3 n - 14\right) \div \left(n + 3\right)$

Divide each polynomial by the binomial.

$\left(u^{3} - 8\right) \div \left(u - 2\right)$

Simplify.

$\left(−5\right)^{−3}$

Simplify.

$\left(6 u\right)^{−3}$

Simplify.

$\left(\frac{3}{4}\right)^{−2}$

Simplify.

$q^{−6} \cdot q^{−5}$

Simplify.

$\left(y^{8}\right)^{−1}$

Simplify.

$\frac{a^{8}}{a^{12}}$

Simplify.

$\frac{r^{−2}}{r^{−3}}$

Write each number in scientific notation.

0.00429

Write each number in scientific notation.

In 2015, the population of the world was about 7,200,000,000 people.

Convert each number to decimal form.

$1.5 \times 10^{10}$

Convert each number to decimal form.

$5.5 \times 10^{−1}$

Multiply and write your answer in decimal form.

$\left(3.5 \times 10^{−2}\right) \left(6.2 \times 10^{−1}\right)$

Divide and write your answer in decimal form.

$\frac{9 \times 10^{−5}}{3 \times 10^{2}}$

Simplify each expression.

$\left(12 a^{2} - 7 a + 4\right) + \left(3 a^{2} + 8 a - 10\right)$

Simplify each expression.

$\left(- \frac{2}{5}\right)^{3}$

Simplify each expression.

$\left(4 a^{3} b^{5}\right)^{2}$

Simplify each expression.

$3 k \left(k^{2} - 7 k + 13\right)$

Simplify each expression.

$\left(v - 9\right) \left(9 v - 5\right)$

Simplify each expression.

$\left(n - 6\right) \left(n^{2} - 5 n + 4\right)$

Simplify each expression.

$\left(7 p - 5\right) \left(7 p + 5\right)$

Simplify each expression.

$\frac{3^{8}}{3^{10}}$

Simplify each expression.

$\left(87 x^{15} y^{3} z^{22}\right)^{0}$

Simplify each expression.

$\frac{12 x^{2} + 42 x - 6}{2 x}$

Simplify each expression.

$\frac{64 x^{3} - 1}{4 x - 1}$

Simplify each expression.

$5^{−2}$

Simplify each expression.

$q^{−4} \cdot q^{−5}$

Simplify each expression.

Convert 83,000,000 to scientific notation.

Simplify, and write your answer in decimal form.

$\left(3.4 \times 10^{9}\right) \left(2.2 \times 10^{−5}\right)$

Simplify, and write your answer in decimal form.

A helicopter flying at an altitude of 1000 feet drops a rescue package. The polynomial $−16 t^{2} + 1000$ gives the height of the package $t$ seconds a after it was dropped. Find the height when $t = 6$ seconds.