5.2 Properties of Exponents and Scientific Notation

Before you get started, take this readiness quiz.

Simplify: $\left(−2\right) \left(−2\right) \left(−2\right) .$
If you missed this problem, review the reference.

Simplify: $\frac{8 x}{24 y} .$
If you missed this problem, review the reference.

Name the decimal $\left(−2.6\right) \left(4.21\right) .$
If you missed this problem, review the reference.

Simplify each expression:

ⓐ $b^{9} \cdot b^{8}$ ⓑ $4^{2 x} \cdot 4^{x}$ ⓒ $3 p^{5} \cdot 4 p$ ⓓ $x^{6} \cdot x^{4} \cdot x^{8} .$

Simplify each expression:

ⓐ $x^{12} \cdot x^{4}$ ⓑ $10 \cdot 10^{x}$ ⓒ $2 z \cdot 6 z^{7}$ ⓓ $b^{5} \cdot b^{9} \cdot b^{5} .$

Simplify each expression: ⓐ $\frac{x^{15}}{x^{10}}$ ⓑ $\frac{6^{14}}{6^{5}}$ ⓒ $\frac{x^{18}}{x^{22}}$ ⓓ $\frac{12^{15}}{12^{30}} .$

Simplify each expression: ⓐ $\frac{y^{43}}{y^{37}}$ ⓑ $\frac{10^{15}}{10^{7}}$ ⓒ $\frac{m^{7}}{m^{15}}$ ⓓ $\frac{9^{8}}{9^{19}} .$

Simplify each expression: ⓐ $9^{0}$ ⓑ $n^{0} .$

Simplify each expression: ⓐ $11^{0}$ ⓑ $q^{0} .$

Simplify each expression: ⓐ $23^{0}$ ⓑ $r^{0} .$

Simplify each expression: ⓐ $x^{−5}$ ⓑ $10^{−3}$ ⓒ $\frac{1}{y^{−4}}$ ⓓ $\frac{1}{3^{−2}} .$

Simplify each expression: ⓐ $z^{−3}$ ⓑ $10^{−7}$ ⓒ $\frac{1}{p^{−8}}$ ⓓ $\frac{1}{4^{−3}} .$

Simplify each expression: ⓐ $n^{−2}$ ⓑ $10^{−4}$ ⓒ $\frac{1}{q^{−7}}$ ⓓ $\frac{1}{2^{−4}} .$

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

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

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

Simplify each expression: ⓐ $z^{−5} \cdot z^{−3}$ ⓑ $\left(m^{4} n^{−3}\right) \left(m^{−5} n^{−2}\right)$ ⓒ $\left(2 x^{−6} y^{8}\right) \left(−5 x^{5} y^{−3}\right) .$

Simplify each expression:

ⓐ $z^{−4} \cdot z^{−5}$ ⓑ $\left(p^{6} q^{−2}\right) \left(p^{−9} q^{−1}\right)$ ⓒ $\left(3 u^{−5} v^{7}\right) \left(−4 u^{4} v^{−2}\right) .$

Simplify each expression:

ⓐ $c^{−8} \cdot c^{−7}$ ⓑ $\left(r^{5} s^{−3}\right) \left(r^{−7} s^{−5}\right)$ ⓒ $\left(−6 c^{−6} d^{4}\right) \left(−5 c^{−2} d^{−1}\right) .$

Simplify each expression: ⓐ $\left(b^{7}\right)^{5}$ ⓑ $\left(5^{4}\right)^{3}$ ⓒ $\left(a^{4}\right)^{5} \left(a^{7}\right)^{4} .$

Simplify each expression: ⓐ $\left(z^{6}\right)^{9}$ ⓑ $\left(3^{7}\right)^{7}$ ⓒ $\left(q^{4}\right)^{5} \left(q^{3}\right)^{3} .$

Simplify each expression: ⓐ $\left(2 w x\right)^{5}$ ⓑ $\left(−11 p q^{3}\right)^{0}$ ⓒ $\left(2 b^{3}\right)^{−4}$ ⓓ $\left(8 a^{−4}\right)^{2} .$

Simplify each expression: ⓐ $\left(−3 y\right)^{3}$ ⓑ $\left(−8 m^{2} n^{3}\right)^{0}$ ⓒ $\left(−4 x^{4}\right)^{−2}$ ⓓ $\left(2 c^{−4}\right)^{3} .$

Simplify each expression:

ⓐ $\left(\frac{p}{10}\right)^{4}$ ⓑ $\left(\frac{m}{n}\right)^{−7}$ ⓒ $\left(\frac{3 a b^{3}}{c^{2}}\right)^{4}$ ⓓ $\left(\frac{3 x^{−2}}{y^{3}}\right)^{3} .$

Simplify each expression:

ⓐ $\left(\frac{−2}{q}\right)^{3}$ ⓑ $\left(\frac{w}{x}\right)^{−4}$ ⓒ $\left(\frac{x y^{3}}{3 z^{2}}\right)^{2}$ ⓓ $\left(\frac{2 m^{−2}}{n^{−2}}\right)^{3} .$

Simplify each expression by applying several properties:

ⓐ $\left(3 x^{2} y\right)^{4} \left(2 x y^{2}\right)^{3}$ ⓑ $\frac{\left(x^{3}\right)^{4} \left(x^{−2}\right)^{5}}{\left(x^{6}\right)^{5}}$ ⓒ $\left(\frac{2 x y^{2}}{x^{3} y^{−2}}\right)^{2} \left(\frac{12 x y^{3}}{x^{3} y^{−1}}\right)^{−1} .$

Simplify each expression:

ⓐ $\left(c^{4} d^{2}\right)^{5} \left(3 c d^{5}\right)^{4}$ ⓑ $\frac{\left(a^{−2}\right)^{3} \left(a^{2}\right)^{4}}{\left(a^{4}\right)^{5}}$ ⓒ $\left(\frac{3 x y^{2}}{x^{2} y^{−3}}\right)^{2} \left(\frac{9 x y^{−3}}{x^{3} y^{2}}\right)^{−1} .$

Simplify each expression:

ⓐ $\left(a^{3} b^{2}\right)^{6} \left(4 a b^{3}\right)^{4}$ ⓑ $\frac{\left(p^{−3}\right)^{4} \left(p^{5}\right)^{3}}{\left(p^{7}\right)^{6}}$ ⓒ $\left(\frac{4 x^{3} y^{2}}{x^{2} y^{−1}}\right)^{2} \left(\frac{8 x y^{−3}}{x^{2} y^{}}\right)^{−1} .$

Write in scientific notation: ⓐ 96,000 ⓑ 0.0078.

Write in scientific notation: ⓐ 48,300 ⓑ 0.0129.

Convert to decimal form: ⓐ $1.3 \times 10^{3}$ ⓑ $−1.2 \times 10^{−4} .$

Convert to decimal form: ⓐ $−9.5 \times 10^{4}$ ⓑ $7.5 \times 10^{−2} .$

Multiply or divide as indicated. Write answers in decimal form: ⓐ $\left(−4 \times 10^{5}\right) \left(2 \times 10^{−7}\right)$ ⓑ $\frac{9 \times 10^{3}}{3 \times 10^{−2}} .$

Multiply or divide as indicated. Write answers in decimal form:

ⓐ $\left(−3 \times 10^{5}\right) \left(2 \times 10^{−8}\right)$ ⓑ $\frac{8 \times 10^{2}}{4 \times 10^{−2}} .$

Multiply or divide as indicated. Write answers in decimal form:

ⓐ $\left(−3 \times 10^{−2}\right) \left(3 \times 10^{−1}\right)$ ⓑ $\frac{8 \times 10^{4}}{2 \times 10^{−1}} .$

Simplify each expression using the properties for exponents.

ⓐ $d^{3} \cdot d^{6}$ ⓑ $4^{5 x} \cdot 4^{9 x}$ ⓒ $2 y \cdot 4 y^{3}$ ⓓ $w \cdot w^{2} \cdot w^{3}$

Simplify each expression using the properties for exponents.

ⓐ $n^{19} \cdot n^{12}$ ⓑ $3^{x} \cdot 3^{6}$ ⓒ $7 w^{5} \cdot 8 w^{}$ ⓓ $a^{4} \cdot a^{3} \cdot a^{9}$

Simplify each expression using the properties for exponents.

$m^{x} \cdot m^{3}$

Simplify each expression using the properties for exponents.

$y^{a} \cdot y^{b}$

Simplify each expression using the properties for exponents.

ⓐ $\frac{x^{18}}{x^{3}}$ ⓑ $\frac{5^{12}}{5^{3}}$ ⓒ $\frac{q^{18}}{q^{36}}$ ⓓ $\frac{10^{2}}{10^{3}}$

Simplify each expression using the properties for exponents.

ⓐ $\frac{p^{21}}{p^{7}}$ ⓑ $\frac{4^{16}}{4^{4}}$ ⓒ $\frac{b}{b^{9}}$ ⓓ $\frac{4}{4^{6}}$

Simplify each expression using the properties for exponents.

ⓐ $20^{0}$ ⓑ $b^{0}$

Simplify each expression using the properties for exponents.

ⓐ $− 27^{0}$ ⓑ $− \left(27^{0}\right)$

Simplify each expression.

ⓐ $a^{−2}$ ⓑ $10^{−3}$ ⓒ $\frac{1}{c^{−5}}$ ⓓ $\frac{1}{3^{−2}}$

Simplify each expression.

ⓐ $r^{−3}$ ⓑ $10^{−5}$ ⓒ $\frac{1}{q^{−10}}$ ⓓ $\frac{1}{10^{−3}}$

Simplify each expression.

ⓐ $\left(\frac{5}{8}\right)^{−2}$ ⓑ $\left(- \frac{b}{a}\right)^{−2}$

Simplify each expression.

ⓐ $\left(\frac{4}{9}\right)^{−3}$ ⓑ $\left(- \frac{u}{v}\right)^{−5}$

Simplify each expression.

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

Simplify each expression.

ⓐ $3 \cdot 5^{−1}$ ⓑ $\left(3 \cdot 5\right)^{−1}$

Simplify each expression using the Product Property.

ⓐ $b^{4} b^{−8}$ ⓑ $\left(w^{4} x^{−5}\right) \left(w^{−2} x^{−4}\right)$ ⓒ $\left(−6 c^{−3} d^{9}\right) \left(2 c^{4} d^{−5}\right)$

Simplify each expression using the Product Property.

ⓐ $a^{3} \cdot a^{−3}$ ⓑ $\left(u v^{−2}\right) \left(u^{−5} v^{−3}\right)$ ⓒ $\left(−4 r^{−2} s^{−8}\right) \left(9 r^{4} s^{3}\right)$

Simplify each expression using the Product Property.

$p^{5} \cdot p^{−2} \cdot p^{−4}$

Simplify each expression using the Power Property.

ⓐ $\left(m^{4}\right)^{2}$ ⓑ $\left(10^{3}\right)^{6}$ ⓒ $\left(x^{3}\right)^{−4}$

Simplify each expression using the Power Property.

ⓐ $\left(y^{3}\right)^{x}$ ⓑ $\left(5^{x}\right)^{y}$ ⓒ $\left(q^{6}\right)^{−8}$

Simplify each expression using the Product to a Power Property.

ⓐ $\left(−3 x y\right)^{2}$ ⓑ $\left(6 a\right)^{0}$ ⓒ $\left(5 x^{2}\right)^{−2}$ ⓓ $\left(−4 y^{−3}\right)^{2}$

Simplify each expression using the Product to a Power Property.

ⓐ $\left(−5 a b\right)^{3}$ ⓑ $\left(−4 p q\right)^{0}$ ⓒ $\left(−6 x^{3}\right)^{−2}$ ⓓ $\left(3 y^{−4}\right)^{2}$

Simplify each expression using the Quotient to a Power Property.

ⓐ $\left(\frac{p}{2}\right)^{5}$ ⓑ $\left(\frac{x}{y}\right)^{−6}$ ⓒ $\left(\frac{2 x y^{2}}{z}\right)^{3}$ ⓓ $\left(\frac{4 p^{−3}}{q^{2}}\right)^{2}$

Simplify each expression using the Quotient to a Power Property.

ⓐ $\left(\frac{a}{3 b}\right)^{4}$ ⓑ $\left(\frac{5}{4 m}\right)^{−2}$ ⓒ $\left(\frac{3 a^{−2} b^{3}}{c^{2}}\right)^{−2}$ ⓓ $\left(\frac{p^{−1} q^{4}}{r^{−4}}\right)^{2}$

Simplify each expression by applying several properties.

ⓐ $\left(5 t^{2}\right)^{3} \left(3 t\right)^{2}$ ⓑ $\frac{\left(t^{2}\right)^{5} \left(t^{−4}\right)^{2}}{\left(t^{3}\right)^{7}}$ ⓒ $\left(\frac{2 x y^{2}}{x^{3} y^{−2}}\right)^{2} \left(\frac{12 x y^{3}}{x^{3} y^{−1}}\right)^{−1}$

Simplify each expression by applying several properties.

ⓐ $\left(m^{2} n\right)^{2} \left(2 m n^{5}\right)^{4}$ ⓑ $\frac{\left(−2 p^{−2}\right)^{4} \left(3 p^{4}\right)^{2}}{\left(−6 p^{3}\right)^{2}}$

Simplify each expression.

ⓐ $7 n^{−1}$ ⓑ $\left(7 n\right)^{−1}$ ⓒ $\left(−7 n\right)^{−1}$

Simplify each expression.

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

Simplify each expression.

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

Simplify each expression.

$\left(a^{2}\right)^{6} \cdot \left(a^{3}\right)^{8}$

Simplify each expression.

$\left(2 m^{6}\right)^{3}$

Simplify each expression.

$\left(10 x^{2} y\right)^{3}$

Simplify each expression.

$\left(−2 a^{3} b^{2}\right)^{4}$

Simplify each expression.

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

Simplify each expression.

$\left(8 a^{3}\right)^{2} \left(2 a\right)^{4}$

Simplify each expression.

$\left(10 p^{4}\right)^{3} \left(5 p^{6}\right)^{2}$

Simplify each expression.

$\left(\frac{1}{2} x^{2} y^{3}\right)^{4} \left(4 x^{5} y^{3}\right)^{2}$

Simplify each expression.

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

Simplify each expression.

ⓐ $\left(3 x\right)^{2} \left(5 x\right)$ ⓑ $\left(2 y\right)^{3} \left(6 y\right)$

Simplify each expression.

ⓐ $\left(2 r^{−2}\right)^{3} \left(4^{−1} r\right)^{2}$ ⓑ $\left(3 x^{−3}\right)^{3} \left(3^{−1} x^{5}\right)^{4}$

Simplify each expression.

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

Simplify each expression.

$\frac{\left(−10 n^{−2}\right)^{3} \left(4 n^{5}\right)^{2}}{\left(2 n^{8}\right)^{2}}$

Write each number in scientific notation.

ⓐ 340,000 ⓑ 0.041

Write each number in scientific notation.

ⓐ 1,290,000 ⓑ 0.00000103

Convert each number to decimal form.

ⓐ $−8.3 \times 10^{2}$ ⓑ $3.8 \times 10^{−2}$

Convert each number to decimal form.

ⓐ $1.6 \times 10^{10}$ ⓑ $8.43 \times 10^{−6}$

Multiply or divide as indicated. Write your answer in decimal form.

ⓐ $\left(2 \times 10^{2}\right) \left(1 \times 10^{−4}\right)$ ⓑ $\frac{5 \times 10^{−2}}{1 \times 10^{−10}}$

Multiply or divide as indicated. Write your answer in decimal form.

ⓐ $\left(3.5 \times 10^{−4}\right) \left(1.6 \times 10^{−2}\right)$ ⓑ $\frac{8 \times 10^{6}}{4 \times 10^{−1}}$

Jennifer thinks the quotient $\frac{a^{24}}{a^{6}}$ simplifies to $a^{4} .$ What is wrong with her reasoning?

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