Exponents

Natural Number Exponents

Natural number exponents are defined by:

\[ x^{n}=x x\cdot\cdot\cdot x\qquad(n factors of x) \]

EXAMPLE 3.1 (a) $ x^{5} = xxxxx $ ; (b) $ 5x^{4}yz{3} = 5xxxzyzz $ ; (c) $ 5a^{3}b + 3(2ab)^{3} = 5aaa b + 3(2ab)(2ab)(2ab) $

Zero as an Exponent

$ x^{0} = 1 $ for x any nonzero real number. $ 0^{0} $ is not defined.

Negative Integer Exponents

Negative integer exponents are defined by:

$ x^{-n} = $ for x any nonzero real number.

$ 0^{-n} $ is not defined for n any positive integer.

EXAMPLE 3.2 (a) $ x^{-5} = $ ; (b) $ 4y^{-3} = 4 = $ ; (c) $ 5^{-3} = = $ ; (d) $ -4^{-2} = - = - $ ;

Rational Number Exponents

$ x^{1/n} $ , the principal nth root of x, is defined, for n an integer greater than 1, by:

If n is odd, $ x^{1/n} $ is the unique real number y which, when raised to the nth power, gives x. If n is even, then,

if \(x > 0\), \(x^{1/n}\) is the positive real number \(y\) which, when raised to the \(n\)th power, gives \(x\).

\[ if x=0,x^{1/n}=0. \]

if x < 0, $ x^{1/n} $ is not a real number.

Note: The principal nth root of a positive number is positive.

EXAMPLE 3.3 (a) $ 8^{1/3} = 2 $ ; (b) $ (-8)^{1/3} = -2 $ ; (c) $ -8^{1/3} = -2 $ ; (d) $ 16^{1/4} = 2 $ ;

5. $ (-16)^{1/4} $ is not a real number; (f) $ -16^{1/4} = -2 $

$ x^{m/n} $ is defined by: $ x^{m/n} = (x^{1/n}){m} $ , provided $ x^{1/n} $ is real.

\[ x^{-m/n}=\frac{1}{x^{m/n}} \]

EXAMPLE 3.4 (a) $ 125^{2/3} = (125^{1/3}){2} = 5^{2} = 25 $ ; (b) $ 8^{-4/3} = = = = $ ;

3. $ (-64)^{5/6} $ is not a real number.

Laws of Exponents

For a and b rational numbers and x and y real numbers (avoiding even roots of negative numbers and division by 0):

\[ x^{a}x^{b}=x^{a+b} \]

\[ (xy)^{a}=x^{a}y^{a}\quad 冚 \quad(x^{a})^{b}=x^{ab} \]

\[ \frac{x^{a}}{x^{b}}=x^{a-b} \]

\[ \frac{x^{a}}{x^{b}}=\frac{1}{x^{b-a}}\qquad\left(\frac{x}{y}\right)^{a}=\frac{x^{a}}{y^{a}} \]

\[ \left(\frac{x}{y}\right)^{-m}=\left(\frac{y}{x}\right)^{m}\qquad\quad\frac{x^{-n}}{y^{-m}}=\frac{y^{m}}{x^{n}} \]

In general, $ x^{m/n} = (x^{1/n}){m} = (x^{m}){1/n} $ , provided $ x^{1/n} $ is real.

Unless otherwise specified, it is generally assumed that variable bases represent positive numbers. With this assumption, then, write $ (x^{n}){1/n} = x $ . However, if this assumption does not hold, then:

$ (x^{n}){1/n}=x $ if n is odd, or if n is even and x is nonnegative

\[ (x^{n})^{1/n}=|x|\qquad if n is even and x is negative \]

EXAMPLE 3.5 If x is known positive: (a) $ (x^{2}){1/2} = x $ ; (b) $ (x^{3}){1/3} = x $ ; (c) $ (x^{4}){1/2} = x^{2} $ ; (d) $ (x^{6}){1/2} = x^{3} $

EXAMPLE 3.6 For general x: (a) $ (x^{2}){1/2} = |x| $ ; (b) $ (x^{3}){1/3} = x $ ; (c) $ (x^{4}){1/2} = |x^{2}| = x^{2} $ ; (d) $ (x^{6}){1/2} = |x^{3}| $

Scientific Notation

In dealing with very large or very small numbers, scientific notation is often used. A number is written in scientific notation when it is expressed as a number between 1 and 10 multiplied by a power of 10.

EXAMPLE 3.7 (a) $ 51,000,000 = 5.1 ^{7} $ ; (b) $ 0.000 000 000 035 2 = 3.52 ^{-11} $ ;

\[ \begin{aligned}(c)\frac{(50,000,000)(0.000\ 000\ 000\ 6)}{(20,000)^{3}}=\frac{(5\times10^{7})(6\times10^{-10})}{(2\times10^{4})^{3}}=\frac{30\times10^{-3}}{8\times10^{12}}=3.75\times10^{-15}\end{aligned} \]

SOLVED PROBLEMS

In the following, bases are assumed to be positive unless otherwise specified:

3.1. Simplify (a) $ 2(3x^{2}y){3}(x^{4}y{3})^{2} $ ; (b) $ $

\[ (a)2(3x^{2}y)^{3}(x^{4}y^{3})^{2}=2\cdot3^{3}x^{6}y^{3}\cdot x^{8}y^{6}=54x^{14}y^{9};(b)\frac{(4x^{5}y^{3})^{2}}{2(xy^{4})^{3}}=\frac{16x^{10}y^{6}}{2x^{3}y^{12}}=\frac{8x^{7}}{y^{6}} \]

3.2. Simplify and write with positive exponents: (a) $ $ ; (b) $ $ ; (c) $ (x^{2} + y^{2}){-2} $ ;

\[ \mathrm{(d)}\ (3x^{-5})^{-2}(5y^{-4})^{3};\ \mathrm{(e)}\ (x^{-2}+y^{-2})^{2};\ \mathrm{(f)}\ \left(\frac{t^{3}u^{4}}{4t^{5}u^{3}}\right)^{-3} \]

\[ \begin{aligned}(a)\frac{x^{2}y^{-3}}{x^{3}y^{3}}&=x^{2-3}y^{-3-3}=x^{-1}y^{-6}=\frac{1}{xy^{6}};(b)\frac{(x^{2}y^{-3})^{-2}}{(x^{3}y^{4})^{-4}}=\frac{x^{-4}y^{6}}{x^{-12}y^{-16}}=x^{-4-(-12)}y^{6-(-16)}=x^{8}y^{22};\end{aligned} \]

\[ \begin{aligned}(c)(x^{2}+y^{2})^{-2}=\frac{1}{(x^{2}+y^{2})^{2}}=\frac{1}{x^{4}+2x^{2}y^{2}+y^{4}};(d)(3x^{-5})^{-2}(5y^{-4})^{3}=3^{-2}x^{10}5^{3}y^{-12}=\frac{125x^{10}}{9y^{12}};\end{aligned} \]

\[ \left(\mathrm{e}\right)(x^{-2}+y^{-2})^{2}=x^{-4}+2x^{-2}y^{-2}+y^{-4}=\frac{1}{x^{4}}+\frac{2}{x^{2}y^{2}}+\frac{1}{y^{4}};\left(\mathrm{f}\right)\left(\frac{t^{3}u^{4}}{4t^{5}u^{3}}\right)^{-3}=\left(\frac{4t^{5}u^{3}}{t^{3}u^{4}}\right)^{3}=\left(\frac{4t^{2}}{u}\right)^{3}=\frac{64t^{6}}{u^{3}} \]

3.3. Simplify: (a) $ x^{1/2}x{1/3} $ ; (b) $ x^{2/3}/x{5/8} $ ; (c) $ (x^{4}y{4})^{-1/2} $ ; (d) $ (x^{4} + y^{4}){-1/2} $

\[ \begin{aligned}(a)x^{1/2}x^{1/3}&=x^{1/2+1/3}=x^{5/6};(b)x^{2/3}/x^{5/8}=x^{2/3-5/8}=x^{1/24};(c)(x^{4}y^{4})^{-1/2}=x^{-2}y^{-2}=\frac{1}{x^{2}y^{2}};\end{aligned} \]

\[ \left(\mathrm{d}\right)(x^{4}+y^{4})^{-1/2}=\frac{1}{(x^{4}+y^{4})^{1/2}} \]

3.4. Simplify: (a) $ 3x^{2/3}y{3/4}(2x^{5/3}y{1/2})^{3} $ ; (b) $ $

\[ \begin{aligned}(a)\quad&3x^{2/3}y^{3/4}(2x^{5/3}y^{1/2})^{3}=3x^{2/3}y^{3/4}\cdot8x^{5}y^{3/2}=24x^{17/3}y^{9/4};\end{aligned} \]

\[ \begin{aligned}(b)\quad\frac{(8x^{2}y^{2/3})^{2/3}}{2(x^{3/4}y)^{3}}=\frac{8^{2/3}x^{4/3}y^{4/9}}{2x^{9/4}y^{3}}=\frac{4x^{4/3}y^{4/9}}{2x^{9/4}y^{3}}=\frac{2}{x^{9/4-4/3}y^{3-4/9}}=\frac{2}{x^{11/12}y^{23/9}}\end{aligned} \]

3.5. Simplify: (a) $ x^{2/3}(x{2} + x + 3) $ ; (b) $ (x^{1/2} + y^{1/2}){2} $ ; (c) $ (x^{1/3} - y^{1/3}){2} $ ; (d) $ (x^{2} + y^{2}){1/2} $

\[ \begin{aligned}(a)\quad&x^{2/3}(x^{2}+x+3)=x^{2/3}x^{2}+x^{2/3}x+3x^{2/3}=x^{8/3}+x^{5/3}+3x^{2/3}\end{aligned} \]

\[ \begin{aligned}(b)\quad&(x^{1/2}+y^{1/2})^{2}=(x^{1/2})^{2}+2x^{1/2}y^{1/2}+(y^{1/2})^{2}=x+2x^{1/2}y^{1/2}+y\end{aligned} \]

\[ \begin{aligned}(c)\quad&(x^{1/3}-y^{1/3})^{2}=(x^{1/3})^{2}-2x^{1/3}y^{1/3}+(y^{1/3})^{2}=x^{2/3}-2x^{1/3}y^{1/3}+y^{2/3}\end{aligned} \]

4. This expression cannot be simplified.

3.6. Factor: (a) $ x^{-4} + 3x^{-2} + 2 $ ; (b) $ x^{2/3} + x^{1/3} - 6 $ ; (c) $ x^{11/3} + 7x^{8/3} + 12x^{5/3} $

1. $ x^{-4} + 3x^{-2} + 2 = (x^{-2} + 1)(x^{-2} + 2) $ using reverse FOIL factoring.

2. $ x^{2/3} + x^{1/3} - 6 = (x^{1/3} + 3)(x^{1/3} - 2) $ using reverse FOIL factoring.

3. $ x^{11/3} + 7x^{8/3} + 12x^{5/3} = x^{5/3}(x{2} + 7x + 12) = x^{5/3}(x + 3)(x + 4) $ removing the monomial common factor, then using reverse FOIL factoring.

3.7. Remove common factors: (a) $ (x + 2)^{-2} + (x + 2)^{-3} $ ; (b) $ 6x^{5}y{-3} - 3y^{-4}x{6} $ ;

\[ \begin{array}{r l}&{\mathrm{(c)~}4(3x+2)^{3}3(x+5)^{-3}-3(x+5)^{-4}(3x+2)^{4};\mathrm{(d)~}5x^{3}(3x+1)^{2/3}+3x^{2}(3x+1)^{5/3}}\end{array} \]

The common factor in such problems, just as in the analogous polynomial problems, consists of each base raised to the smallest exponent present in each term.

\[ \begin{aligned}(a)\quad&(x+2)^{-2}+(x+2)^{-3}=(x+2)^{-3}[(x+2)^{-2-(-3)}+1]=(x+2)^{-3}(x+2+1)=(x+2)^{-3}(x+3)\end{aligned} \]

\[ \begin{aligned}(b)\quad&6x^{5}y^{-3}-3y^{-4}x^{6}=3x^{5}y^{-4}(2y^{-3-(-4)}-x^{6-5})=3x^{5}y^{-4}(2y-x)\end{aligned} \]

\[ \begin{align*}(c)\ 4(3x+2)^{3}3(x+5)^{-3}-3(x+5)^{-4}(3x+2)^{4}&=3(3x+2)^{3}(x+5)^{-4}[4(x+5)-(3x+2)]\\&=3(3x+2)^{3}(x+5)^{-4}(x+18)\end{align*} \]

\[ \begin{aligned}(d)\ 5x^{3}(3x+1)^{2/3}+3x^{2}(3x+1)^{5/3}&=x^{2}(3x+1)^{2/3}[5x+3(3x+1)^{5/3-2/3}]\\&=x^{2}(3x+1)^{2/3}[5x+3(3x+1)]\\&=x^{2}(3x+1)^{2/3}(14x+3)\end{aligned} \]

3.8. Simplify: (a) $ $ ; (b) $ (x^{p+1}){2}(x^{p-1}){2} $ ; (c) $ ()^{1/n} $

\[ \begin{aligned}(a)\quad\frac{x^{p+q}}{x^{p-q}}=x^{(p+q)-(p-q)}=x^{p+q-p+q}=x^{2q}\end{aligned} \]

\[ \begin{array}{r l}{(\mathbf{b})}&{(x^{p+1})^{2}(x^{p-1})^{2}=x^{2(p+1)}x^{2(p-1)}=x^{(2p+2)+(2p-2)}=x^{4p}}\end{array} \]

\[ \left(\mathrm{c}\right)\quad\left(\frac{x^{mn}}{x^{n^{2}}}\right)^{1/n}=\frac{x^{mn(1/n)}}{x^{n^{2}(1/n)}}=\frac{x^{m}}{x^{n}}=x^{m-n} \]

3.9. Simplify, without assuming that variable bases are positive:

1. \((x^{4})^{1/4}\); (b) \((x^{2}y^{4}z^{6})^{1/2}\); (c) \((x^{3}y^{6}z^{9})^{1/3}\); (d) \([x(x+h)^{2}]^{1/2}\)

\[ \begin{array}{r l}&{\mathrm{(a)~}(x^{4})^{1/4}=\mathrm{l x l};\mathrm{(b)~}(x^{2}y^{4}z^{6})^{1/2}=(x^{2})^{1/2}(y^{4})^{1/2}(z^{6})^{1/2}=\mathrm{l x l}\mathrm{|y^{2}|}\mathrm{|z^{3}|}=\mathrm{|x|y^{2}|z^{3}|};\mathrm{(c)~}(x^{3}y^{6}z^{9})^{1/3}=(x^{3})^{1/3}(y^{6})^{1/3}(z^{9})^{1/3}=x y^{2}z^{3};}\end{array} \]

4. $ [x(x + h)^{2}]^{1/2} = x^{1/2}[(x + h)^{2}]^{1/2} = x^{1/2}|x + h| $

3.10. (a) Write in scientific notation: The velocity of light is 186,000 mi/sec. (b) Find the number of seconds in a year and write the answer in scientific notation. (c) Express the distance light travels in 1 year in scientific notation.

1. Moving the decimal point to the right of the first nonzero digit is a shift of 5 places: thus, 186,000 miles/sec = 1.86 × 10^{5} mi/sec.

2. 1 year = 365 days × 24 hours/day × 60 minutes/hour × 60 seconds/minute

$ = 31.536.000 = 3.15 ^{7} $ .

3. Since distance = velocity × time, the distance light travels in 1 year

\[ =(1.86\times10^{5}mi/sec)\times(3.15\times10^{7}sec)=5.87\times10^{12}mi. \]

SUPPLEMENTARY PROBLEMS

3.11. Simplify:(a) $ (xy^{3}){4}(3x^{2}y){3} $ ; (b) $ $

Ans. (a) $ 27x^{10}y{15} $ ; (b) $ $

3.12. Simplify: (a) $ 2(xy^{-3}){-2}(4x^{-3}y{2})^{-1} $ ; (b) $ ()^{-2} $

Ans. (a) $ $ ; (b) $ $

3.13. Simplify, assuming all variable bases are positive: (a) $ (8y^{3}z{4})^{2/3} $ ; (b) $ (100x^{8}y{3})^{-1/2} $ ; (c) $ ()^{-2/3} $ ; (d) $ ()^{-3/2} $

Ans. (a) $ 4y^{2}z{8/3} $ ; (b) $ $ ; (c) $ $ ; (d) $ $

3.14. Simplify, assuming all variable bases are positive:

1. \((x^{4}y^{3})^{1/2}(8x^{6}y)^{2/3}\); (b) \((9x^{8}y)^{-1/2}(16x^{-4}y^{3})^{3/2}\)

Ans. (a) $ 4x^{6}y{13/6} $ ; (b) $ $

3.15. Calculate: (a) $ 25^{-1/2} - 16^{-1/2} $ ; (b) $ (25 - 16)^{-1/2} $ ; (c) $ 16^{3/4} + 16^{-3/4} $

Ans. (a) $ - $ ; (b) $ $ ; (c) $ $

3.16. Simplify: (a) $ x^{0} + y^{0} + (x + y)^{0} $ ; (b) $ ()^{-2} $ ; (c) $ ()^{3/5} $ ; (d) $ $

Ans. (a) 3; (b) $ $ ; (c) $ $ ; (d) $ $

3.17. Derive the laws $ = $ and $ ()^{-m} = ()^{m} $ from the definition of negative exponents and standard fraction operations.

3.18. Perform indicated operations: (a) $ (x^{1/2} + y^{1/2})(x{1/2} - y^{1/2}) $ ; (b) $ (x^{1/3} + y^{1/3})(x{1/3} - y^{1/3}) $ ;

\[ \begin{array}{r l}&{(\mathbf{c})(x^{1/3}+y^{1/3})(x^{2/3}-x^{1/3}y^{1/3}+y^{2/3});(\mathbf{d})(x^{1/3}+y^{1/3})(x^{2/3}+x^{1/3}y^{1/3}+y^{2/3});(\mathbf{e})(x^{2/3}-y^{2/3})^{3}\end{array} \]

\[ Ans.\quad(a)x-y;(b)x^{2/3}-y^{2/3};(c)x+y;(d)x+2x^{2/3}y^{1/3}+2x^{1/3}y^{2/3}+y;(e)x^{2}-3x^{4/3}y^{2/3}+3x^{2/3}y^{4/3}-y^{2} \]

3.19. Remove common factors: (a) $ x^{-8}y{-7} + x^{-7}y{-8} $ ; (b) $ x^{-5/3}y{3} - x^{-2/3}y{2} $ ; (c) $ x^{p+q} + x^{p} $ ;

\[ \left(\mathrm{d}\right)4(x^{2}+4)^{3/2}(3x+5)^{1/3}+(3x+5)^{4/3}(x^{2}+4)^{1/2}3x \]

\[ Ans.\quad(a)x^{-8}y^{-8}(y+x);(b)x^{-5/3}y^{2}(y-x);(c)x^{p}(x^{q}+1);(d)(3x+5)^{1/3}(x^{2}+4)^{1/2}(13x^{2}+15x+16) \]

3.20. Remove common factors: (a) $ x^{-5} + 2x^{-4} + 2x^{-3} $ ; (b) $ 6x^{2}(x{2} - 1)^{3/2} + x^{3}(x{2} - 1)^{1/2}(6x) $ ;

\[ \begin{aligned}(c)-4x^{-5}(1-x^{2})^{3}+x^{-4}(6x)(1-x^{2})^{2};(d)x^{-4}(1-2x)^{-3/2}-4x^{-5}(1-2x)^{-1/2}\end{aligned} \]

\[ Ans.\quad(a)x^{-5}(1+2x+2x^{2});(b)6x^{2}(x^{2}-1)^{1/2}(2x^{2}-1);(c)2x^{-5}(1-x^{2})^{2}(5x^{2}-2);(d)x^{-5}(1-2x)^{-3/2}(9x-4) \]

3.21. Remove common factors:

\[ (a)\ -(x-2)^{-2}(3x-7)^{-3}-3(x-2)^{-1}(3x-7)^{-4}(3); \]

\[ \begin{aligned}(b)-4(x^{2}-4)^{-5}(2x)(x^{2}+4)^{3}+3(x^{2}-4)^{-4}(x^{2}+4)^{2}(2x)\end{aligned} \]

\[ Ans.\quad(a)\ -(x-2)^{-2}(3x-7)^{-4}(12x-25);\\ (b)\ (x^{2}-4)^{-5}(x^{2}+4)^{2}(2x)(-x^{2}-28) \]

3.22. Remove common factors:

\[ (a)3(x+3)^{2}(3x-1)^{1/2}+(x+3)^{3}\Big(\frac{1}{2}\Big)(3x-1)^{-1/2}(3); \]

\[ (b)\frac{3}{2}(2x+3)^{1/2}(3x+4)^{4/3}(2)+(2x+3)^{3/2}\Big(\frac{4}{3}\Big)(3x+4)^{1/3}(3); \]

\[ (c)\ -\frac{3}{2}(4x^{2}\ -\ 1)^{-5/2}(8x)(1\ +\ x^{2})^{2/3}\ +\ (4x^{2}\ -\ 1)^{-3/2}\Big(\frac{2}{3}\Big)(1\ +\ x^{2})^{-1/3}(2x) \]

\[ Ans.\quad(a)\frac{3}{2}(x+3)^{2}(3x-1)^{-1/2}(7x+1);(b)(2x+3)^{1/2}(3x+4)^{1/3}(17x+24); \]

\[ \left(\mathrm{c}\right)\frac{4}{3}x(4x^{2}-1)^{-5/2}(1+x^{2})^{-1/3}(-5x^{2}-10) \]

3.23. Simplify and write in scientific notation: (a) $ (7.2 ^{-3})(5 ^{12}) $ ;

\[ (b)(7.2\times10^{-3})\div(5\times10^{12});(c)\frac{(3\times10^{-5})(6\times10^{-3})^{3}}{(9\times10^{-12})^{2}} \]

\[ Ans.\quad(a)3.6\times10^{10};(b)1.44\times10^{-15};(c)8\times10^{10} \]

3.24. There are approximately $ 6.01 ^{23} $ atoms of hydrogen in one gram. Calculate the approximate mass in grams of one hydrogen atom.

Ans. $ 1.67 ^{-24} $ grams

3.25. According to the United States Department of Commerce, the U.S. Gross Domestic Product (GDP) for 2017 was $19,390,000,000,000. According to the United States Bureau of the Census, the U.S. population was 325,700,000 (January 2017). Write these figures in scientific notation and use the result to estimate the GDP per person as of 2017 in scientific notation and in standard notation.

\[ Ans.\quad1.939\times10^{13},3.257\times10^{8},5.953\times10^{4}or\59,530. \]

3.26. “In March 2017 the federal debt limit was set at $19,808,800,000,000. Write this figure in scientific notation and use this to estimate each U.S. inhabitant’s share of the debt, in scientific notation and in standard notation.

\[ Ans.\quad1.98088\times10^{13},6.0816\times10^{4}or\60,816. \]