# 7.3 Double-angle, half-angle, and reduction formulas  (Page 5/8)

 Page 5 / 8

If $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x=\frac{2}{3},$ and $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is in quadrant I.

If $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x=-\frac{1}{2},$ and $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is in quadrant III.

a) $\text{\hspace{0.17em}}\frac{\sqrt{3}}{2}\text{\hspace{0.17em}}$ b) $\text{\hspace{0.17em}}-\frac{1}{2}\text{\hspace{0.17em}}$ c) $\text{\hspace{0.17em}}-\sqrt{3}\text{\hspace{0.17em}}$

If $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x=-8,$ and $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is in quadrant IV.

For the following exercises, find the values of the six trigonometric functions if the conditions provided hold.

$\mathrm{cos}\left(2\theta \right)=\frac{3}{5}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{90}^{\circ }\le \theta \le {180}^{\circ }$

$\mathrm{cos}\text{\hspace{0.17em}}\theta =-\frac{2\sqrt{5}}{5},\mathrm{sin}\text{\hspace{0.17em}}\theta =\frac{\sqrt{5}}{5},\mathrm{tan}\text{\hspace{0.17em}}\theta =-\frac{1}{2},\mathrm{csc}\text{\hspace{0.17em}}\theta =\sqrt{5},\mathrm{sec}\text{\hspace{0.17em}}\theta =-\frac{\sqrt{5}}{2},\mathrm{cot}\text{\hspace{0.17em}}\theta =-2$

$\mathrm{cos}\left(2\theta \right)=\frac{1}{\sqrt{2}}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{180}^{\circ }\le \theta \le {270}^{\circ }$

For the following exercises, simplify to one trigonometric expression.

$2\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\pi }{4}\right)\text{\hspace{0.17em}}2\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\pi }{4}\right)$

$2\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\pi }{2}\right)$

$4\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\pi }{8}\right)\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\pi }{8}\right)$

For the following exercises, find the exact value using half-angle formulas.

$\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\pi }{8}\right)\text{\hspace{0.17em}}$

$\frac{\sqrt{2-\sqrt{2}}}{2}$

$\mathrm{cos}\left(-\frac{11\pi }{12}\right)$

$\mathrm{sin}\left(\frac{11\pi }{12}\right)$

$\frac{\sqrt{2-\sqrt{3}}}{2}$

$\mathrm{cos}\left(\frac{7\pi }{8}\right)$

$\mathrm{tan}\left(\frac{5\pi }{12}\right)$

$2+\sqrt{3}$

$\mathrm{tan}\left(-\frac{3\pi }{12}\right)$

$\mathrm{tan}\left(-\frac{3\pi }{8}\right)$

$-1-\sqrt{2}$

For the following exercises, find the exact values of a) $\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{x}{2}\right),$ b) $\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{x}{2}\right),$ and c) $\text{\hspace{0.17em}}\mathrm{tan}\left(\frac{x}{2}\right)$ without solving for $\text{\hspace{0.17em}}x.$

If $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x=-\frac{4}{3},$ and $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is in quadrant IV.

If $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x=-\frac{12}{13},$ and $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is in quadrant III.

a) $\text{\hspace{0.17em}}\frac{3\sqrt{13}}{13}\text{\hspace{0.17em}}$ b) $\text{\hspace{0.17em}}-\frac{2\sqrt{13}}{13}\text{\hspace{0.17em}}$ c) $\text{\hspace{0.17em}}-\frac{3}{2}\text{\hspace{0.17em}}$

If $\text{\hspace{0.17em}}\mathrm{csc}\text{\hspace{0.17em}}x=7,$ and $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is in quadrant II.

If $\text{\hspace{0.17em}}\mathrm{sec}\text{\hspace{0.17em}}x=-4,$ and $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is in quadrant II.

a) $\text{\hspace{0.17em}}\frac{\sqrt{10}}{4}\text{\hspace{0.17em}}$ b) $\text{\hspace{0.17em}}\frac{\sqrt{6}}{4}\text{\hspace{0.17em}}$ c) $\text{\hspace{0.17em}}\frac{\sqrt{15}}{3}\text{\hspace{0.17em}}$

For the following exercises, use [link] to find the requested half and double angles.

Find $\text{\hspace{0.17em}}\mathrm{sin}\left(2\theta \right),\mathrm{cos}\left(2\theta \right),$ and $\text{\hspace{0.17em}}\mathrm{tan}\left(2\theta \right).$

Find $\text{\hspace{0.17em}}\mathrm{sin}\left(2\alpha \right),\mathrm{cos}\left(2\alpha \right),$ and $\text{\hspace{0.17em}}\mathrm{tan}\left(2\alpha \right).$

$\frac{120}{169},–\frac{119}{169},–\frac{120}{119}$

Find $\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\theta }{2}\right),\mathrm{cos}\left(\frac{\theta }{2}\right),$ and $\text{\hspace{0.17em}}\mathrm{tan}\left(\frac{\theta }{2}\right).$

Find $\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\alpha }{2}\right),\mathrm{cos}\left(\frac{\alpha }{2}\right),$ and $\text{\hspace{0.17em}}\mathrm{tan}\left(\frac{\alpha }{2}\right).$

$\frac{2\sqrt{13}}{13},\frac{3\sqrt{13}}{13},\frac{2}{3}$

For the following exercises, simplify each expression. Do not evaluate.

${\mathrm{cos}}^{2}\left({28}^{\circ }\right)-{\mathrm{sin}}^{2}\left({28}^{\circ }\right)$

$2{\mathrm{cos}}^{2}\left({37}^{\circ }\right)-1$

$\mathrm{cos}\left({74}^{\circ }\right)$

$1-2{\text{\hspace{0.17em}}\mathrm{sin}}^{2}\left({17}^{\circ }\right)$

${\mathrm{cos}}^{2}\left(9x\right)-{\mathrm{sin}}^{2}\left(9x\right)$

$\mathrm{cos}\left(18x\right)$

$4\text{\hspace{0.17em}}\mathrm{sin}\left(8x\right)\text{\hspace{0.17em}}\mathrm{cos}\left(8x\right)$

$6\text{\hspace{0.17em}}\mathrm{sin}\left(5x\right)\text{\hspace{0.17em}}\mathrm{cos}\left(5x\right)$

$3\mathrm{sin}\left(10x\right)$

For the following exercises, prove the identity given.

${\left(\mathrm{sin}\text{\hspace{0.17em}}t-\mathrm{cos}\text{\hspace{0.17em}}t\right)}^{2}=1-\mathrm{sin}\left(2t\right)$

$\mathrm{sin}\left(2x\right)=-2\text{\hspace{0.17em}}\mathrm{sin}\left(-x\right)\text{\hspace{0.17em}}\mathrm{cos}\left(-x\right)$

$-2\text{\hspace{0.17em}}\mathrm{sin}\left(-x\right)\mathrm{cos}\left(-x\right)=-2\left(-\mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)\right)=\mathrm{sin}\left(2x\right)$

$\mathrm{cot}\text{\hspace{0.17em}}x-\mathrm{tan}\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}\mathrm{cot}\left(2x\right)$

$\frac{\mathrm{sin}\left(2\theta \right)}{1+\mathrm{cos}\left(2\theta \right)}{\mathrm{tan}}^{2}\theta =\mathrm{tan}\text{\hspace{0.17em}}\theta$

$\begin{array}{l}\frac{\mathrm{sin}\left(2\theta \right)}{1+\mathrm{cos}\left(2\theta \right)}{\mathrm{tan}}^{2}\theta =\frac{2\mathrm{sin}\left(\theta \right)\mathrm{cos}\left(\theta \right)}{1+{\mathrm{cos}}^{2}\theta -{\mathrm{sin}}^{2}\theta }{\mathrm{tan}}^{2}\theta =\\ \frac{2\mathrm{sin}\left(\theta \right)\mathrm{cos}\left(\theta \right)}{2{\mathrm{cos}}^{2}\theta }{\mathrm{tan}}^{2}\theta =\frac{\mathrm{sin}\left(\theta \right)}{\mathrm{cos}\text{\hspace{0.17em}}\theta }{\mathrm{tan}}^{2}\theta =\\ \mathrm{cot}\left(\theta \right){\mathrm{tan}}^{2}\theta =\mathrm{tan}\text{\hspace{0.17em}}\theta \end{array}$

For the following exercises, rewrite the expression with an exponent no higher than 1.

${\mathrm{cos}}^{2}\left(5x\right)$

${\mathrm{cos}}^{2}\left(6x\right)$

$\frac{1+\mathrm{cos}\left(12x\right)}{2}$

${\mathrm{sin}}^{4}\left(8x\right)$

${\mathrm{sin}}^{4}\left(3x\right)$

$\frac{3+\mathrm{cos}\left(12x\right)-4\mathrm{cos}\left(6x\right)}{8}$

${\mathrm{cos}}^{2}x{\text{\hspace{0.17em}}\mathrm{sin}}^{4}x$

${\mathrm{cos}}^{4}x{\text{\hspace{0.17em}}\mathrm{sin}}^{2}x$

$\frac{2+\mathrm{cos}\left(2x\right)-2\mathrm{cos}\left(4x\right)-\mathrm{cos}\left(6x\right)}{32}$

${\mathrm{tan}}^{2}x{\text{\hspace{0.17em}}\mathrm{sin}}^{2}x$

## Technology

For the following exercises, reduce the equations to powers of one, and then check the answer graphically.

${\mathrm{tan}}^{4}x$

$\frac{3+\mathrm{cos}\left(4x\right)-4\mathrm{cos}\left(2x\right)}{3+\mathrm{cos}\left(4x\right)+4\mathrm{cos}\left(2x\right)}$

${\mathrm{sin}}^{2}\left(2x\right)$

${\mathrm{sin}}^{2}x{\text{\hspace{0.17em}}\mathrm{cos}}^{2}x$

$\frac{1-\mathrm{cos}\left(4x\right)}{8}$

${\mathrm{tan}}^{2}x\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x$

${\mathrm{tan}}^{4}x{\text{\hspace{0.17em}}\mathrm{cos}}^{2}x$

$\frac{3+\mathrm{cos}\left(4x\right)-4\mathrm{cos}\left(2x\right)}{4\left(\mathrm{cos}\left(2x\right)+1\right)}$

${\mathrm{cos}}^{2}x\text{\hspace{0.17em}}\mathrm{sin}\left(2x\right)$

${\mathrm{cos}}^{2}\left(2x\right)\mathrm{sin}\text{\hspace{0.17em}}x$

$\frac{\left(1+\mathrm{cos}\left(4x\right)\right)\mathrm{sin}\text{\hspace{0.17em}}x}{2}$

${\mathrm{tan}}^{2}\left(\frac{x}{2}\right)\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x$

For the following exercises, algebraically find an equivalent function, only in terms of $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and/or $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x,$ and then check the answer by graphing both equations.

$\mathrm{sin}\left(4x\right)$

$4\mathrm{sin}\text{\hspace{0.17em}}x\mathrm{cos}\text{\hspace{0.17em}}x\left({\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x\right)$

$\mathrm{cos}\left(4x\right)$

## Extensions

For the following exercises, prove the identities.

$\mathrm{sin}\left(2x\right)=\frac{2\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x}{1+{\mathrm{tan}}^{2}x}$

$\frac{2\mathrm{tan}\text{\hspace{0.17em}}x}{1+{\mathrm{tan}}^{2}x}=\frac{\frac{2\mathrm{sin}\text{\hspace{0.17em}}x}{\mathrm{cos}\text{\hspace{0.17em}}x}}{1+\frac{{\mathrm{sin}}^{2}x}{{\mathrm{cos}}^{2}x}}=\frac{\frac{2\mathrm{sin}\text{\hspace{0.17em}}x}{\mathrm{cos}\text{\hspace{0.17em}}x}}{\frac{{\mathrm{cos}}^{2}x+{\mathrm{sin}}^{2}x}{{\mathrm{cos}}^{2}x}}=$
$\frac{2\mathrm{sin}\phantom{\rule{0.2em}{0ex}}x}{\mathrm{cos}\phantom{\rule{0.2em}{0ex}}x}.\frac{{\mathrm{cos}}^{2}x}{1}=2\mathrm{sin}x\mathrm{cos}\phantom{\rule{0.2em}{0ex}}x=\mathrm{sin}\left(2x\right)$

$\mathrm{cos}\left(2\alpha \right)=\frac{1-{\mathrm{tan}}^{2}\alpha }{1+{\mathrm{tan}}^{2}\alpha }$

$\mathrm{tan}\left(2x\right)=\frac{2\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x\mathrm{cos}\text{\hspace{0.17em}}x}{2{\mathrm{cos}}^{2}x-1}$

$\frac{2\mathrm{sin}\text{\hspace{0.17em}}x\mathrm{cos}\text{\hspace{0.17em}}x}{2{\mathrm{cos}}^{2}x-1}=\frac{\mathrm{sin}\left(2x\right)}{\mathrm{cos}\left(2x\right)}=\mathrm{tan}\left(2x\right)$

${\left({\mathrm{sin}}^{2}x-1\right)}^{2}=\mathrm{cos}\left(2x\right)+{\mathrm{sin}}^{4}x$

$\mathrm{sin}\left(3x\right)=3\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x-{\mathrm{sin}}^{3}x$

$\begin{array}{l}\mathrm{sin}\left(x+2x\right)=\mathrm{sin}\text{\hspace{0.17em}}x\mathrm{cos}\left(2x\right)+\mathrm{sin}\left(2x\right)\mathrm{cos}\text{\hspace{0.17em}}x\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\mathrm{sin}\text{\hspace{0.17em}}x\left({\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x\right)+2\mathrm{sin}\text{\hspace{0.17em}}x\mathrm{cos}\text{\hspace{0.17em}}x\mathrm{cos}\text{\hspace{0.17em}}x\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\mathrm{sin}\text{\hspace{0.17em}}x{\mathrm{cos}}^{2}x-{\mathrm{sin}}^{3}x+2\mathrm{sin}\text{\hspace{0.17em}}x{\mathrm{cos}}^{2}x\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=3\mathrm{sin}\text{\hspace{0.17em}}x{\mathrm{cos}}^{2}x-{\mathrm{sin}}^{3}x\hfill \end{array}$

$\mathrm{cos}\left(3x\right)={\mathrm{cos}}^{3}x-3{\mathrm{sin}}^{2}x\mathrm{cos}\text{\hspace{0.17em}}x$

$\frac{1+\mathrm{cos}\left(2t\right)}{\mathrm{sin}\left(2t\right)-\mathrm{cos}\text{\hspace{0.17em}}t}=\frac{2\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}t}{2\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}t-1}$

$\begin{array}{l}\frac{1+\mathrm{cos}\left(2t\right)}{\mathrm{sin}\left(2t\right)-\mathrm{cos}t}=\frac{1+2{\mathrm{cos}}^{2}t-1}{2\mathrm{sin}t\mathrm{cos}t-\mathrm{cos}t}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{2{\mathrm{cos}}^{2}t}{\mathrm{cos}t\left(2\mathrm{sin}t-1\right)}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{2\mathrm{cos}t}{2\mathrm{sin}t-1}\hfill \end{array}$

$\mathrm{sin}\left(16x\right)=16\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{cos}\left(2x\right)\mathrm{cos}\left(4x\right)\mathrm{cos}\left(8x\right)$

$\mathrm{cos}\left(16x\right)=\left({\mathrm{cos}}^{2}\left(4x\right)-{\mathrm{sin}}^{2}\left(4x\right)-\mathrm{sin}\left(8x\right)\right)\left({\mathrm{cos}}^{2}\left(4x\right)-{\mathrm{sin}}^{2}\left(4x\right)+\mathrm{sin}\left(8x\right)\right)$

how fast can i understand functions without much difficulty
what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this