# 1.2 Exponents and scientific notation  (Page 5/9)

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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. ${\left({c}^{-5}{d}^{-3}\right)}^{4}$
1. $\frac{{b}^{15}}{{c}^{3}}$
2. $\frac{625}{{u}^{32}}$
3. $\frac{-1}{{w}^{105}}$
4. $\frac{{q}^{24}}{{p}^{32}}$
5. $\frac{1}{{c}^{20}{d}^{12}}$

## Simplifying exponential expressions

Recall that to simplify an expression means to rewrite it by combing terms or exponents; in other words, to write the expression more simply with fewer terms. The rules for exponents may be combined to simplify expressions.

## Simplifying exponential expressions

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. ${\left(2u{v}^{-2}\right)}^{-3}$
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. $\frac{{\left(2{h}^{2}k\right)}^{4}}{{\left(7{h}^{-1}{k}^{2}\right)}^{2}}$
1. $\frac{{v}^{6}}{8{u}^{3}}$
2. $\frac{1}{{x}^{3}}$
3. $\frac{{e}^{4}}{{f}^{4}}$
4. $\frac{27r}{s}$
5. $1$
6. $\frac{16{h}^{10}}{49}$

## Using scientific notation

Recall at the beginning of the section that we found the number $\text{\hspace{0.17em}}1.3\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{13}\text{\hspace{0.17em}}$ when describing bits of information in digital images. Other extreme numbers include the width of a human hair, which is about 0.00005 m, and the radius of an electron, which is about 0.00000000000047 m. How can we effectively work read, compare, and calculate with numbers such as these?

A shorthand method of writing very small and very large numbers is called scientific notation    , in which we express numbers in terms of exponents of 10. To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between 1 and 10. Count the number of places n that you moved the decimal point. Multiply the decimal number by 10 raised to a power of n . If you moved the decimal left as in a very large number, $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is positive. If you moved the decimal right as in a small large number, $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is negative.

For example, consider the number 2,780,418. Move the decimal left until it is to the right of the first nonzero digit, which is 2.

We obtain 2.780418 by moving the decimal point 6 places to the left. Therefore, the exponent of 10 is 6, and it is positive because we moved the decimal point to the left. This is what we should expect for a large number.

$2.780418\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{6}$

Working with small numbers is similar. Take, for example, the radius of an electron, 0.00000000000047 m. Perform the same series of steps as above, except move the decimal point to the right.

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
More example of algebra and trigo
What is Indices
If one side only of a triangle is given is it possible to solve for the unkown two sides?
cool
Rubben
kya
Khushnama
please I need help in maths
Okey tell me, what's your problem is?
Navin