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No you can simplify but you can remove de root from the denominator

That if you want to delete the root from the denominator it is the same.
Good Luck, glad to help you

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That polynomial can be expressed as:
(x-1)(x-1)(x-3) Now we have to expand this expression
(x-1)(x-1)(x-3)=(x^2-2x+1)(x-3)=x^3-2x^2+x-3x^2+6x-3=x^3-5x^2+7x-3
So the polynomial of degree three that have the solution (1,1,3) is
x^3-5x^2+7x-3
I'm glad to help you anytime

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We substarct the real numbers and the complex number apart:
(-2 -6i) - (6 - i) =( -8 -5i)

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It is an open interval. Then we have to use the following simbols >to expres greatter than, and< to express less than.
Then:

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The logarithm of a product of two number is the sum of the logarithms of both numbers:
log(xy)=log(x)+log(y)
and otherwise(similary):
log(x/y)=log(x)-log(y)
Then:
log(xy^2)=log(x)+log(y^2)

log(xy^2)/log(y)=log(x)+log(y^2)-log(y)

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The slope form is:
y=mx+b
m is the slope and b is the y-intercept.
From the data we know that when x=3, y=-2
Then:
-2=3m+b
If the slope if five we have:
-2=3(5)+b
-2=15+b
-17=b
Finally we get the slope form of the line as follow:

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True

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Data:
v = 331m/s speed of sound
t = 12 second time take by the sound to arrive to Joe
d = x distance from Joe to the lightning flash.
Above we have the data that will be used to find the distance.
Formula:

This is the formula to find the distance given the speed and the time.
Using the data we have:


Answer: The lightning flash is 3972 meters away from Joe.

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Let call the angle X
Its complement will be 90-x
Its supplement will be 180-x
(180-x)+15 = 4(90-x)
180-x+15 = 360-4x
4x-x = 360-180-15
3x = 165
x = 55
Prove
The complement of 55 is 35
And the supplement of 55 is 125
Now we must see if the relation stated holds
125+15 = 4(35)
125+15 = 140 The measure of the supplement of an angle is 15 degrees less than four times the measure of the complement.
140 = 140
Therefore the solution of the problem is given by an angle of 55 degrees.

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Perimeter P=130 feet
Width w=x
Length l=4x
Perimeter P=2(w+l)
Applying the fromula
P = 2(x+4x)
P = 2(5x)
P = 10x
130 = 10x
130/10 = x
13 = x ,therfore the width is 13 feet

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2/4=1/2 Dived by the common divisor or commmon factor.
4/5 Can not be simplified because 4 and 5 have not common factors.

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Is also referred as the Euler's number.

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The perimeter is describe as the sum of the measures of all sides in a geometrical figure.
Then:

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8.38 x 10^7 feet - 5.84 x 10^7 feet = 2.54 x 10^7 feet
Explanation:
If the exponent of ten are equals(in this case 7) we only have to substract the coefficients. Generally:

With the sums is the same:

With multiplication and division change a little:
For multiplication
For division

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Given a number x, the reciprocal of x is 1/x.
6 times the reciprocal of x, is 6/x
and the problem says that the sum of x and its reciprocal is equal to -5,then:
x+6/x=-5 Now we have to eliminate the denominator by multiplying the entire expression by x
x(x)+x(6/x)=x(-5)
x^2+6=-5x
x^2+5x+6=0 Now we have to solve the equation to solve the problem and find the solutions.
The solution is given using the general formula for quadratic equations:

Solutions: x1=-2,x2=-3
We have to solutions and therefore two pair of number for which the stated condition hold.
Those numbers are:
-2 and -1/2 , -3 and -1/3
Proof:
-2+6(-1/2) = -2-3=-5
-3+6(-1/3) = -3-2=-5
Then the solutions are true.
Any doubt?:)

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log (x^3y/z^2) = 3log(x)+log(y)-2log(z)
Explanation
The logarithm of the product of two number is equal to the sum of the logarithm of each number.
log(xy)=log(x)+log(y)
The logarithm of the division of two numbers is equal to the substraction of the logarithm of each number.
log(x/y)=log(x)-log(y)
Remember that the base does not change. You may use this properties to solve any problem like the above.

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Solution
1) log(base10)7x^4
log(7)+4log(x), remember log in base 10
2) log(base7)x^1/2 / x
1/2log(x)-log(x), know in base 7
3) ln 3xy^2
Ln(3)+Ln(x)+2Ln(y), I suppose that this is in base e

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3 ln x + 2 ln (x+1) = ln x^3+ln(x+1)^2 = ln (x^3(x+1)^2
If you need something more please let me know.

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We can solve this system by substitution:
x-y=3, From this equation we get x=3+y
2x-y=-1, Substituting in the second equation we get 2(3+y)-y=-1,
solving 6+2y-y=-1, y=-1-6, y=-7,
And replacing the obtained value in the equation x=3+y, we obtain x=3+(-7),
x=-4
The solutions to the system are x=-4, y=-7.

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System.
2x+4y=10
3x-2y=-9
We can easily solve this system by reduction:
2x+4y=10
2(3x-2y)=2(-9) We multiply the second equation by two getting a similar equation.
2x+4y=10
6x-4y=-18 We have the same system, now we only have to sum the corresponding parts.
2x+4y=10
+6x-4y=-18
Result:8x =-8
Solving for x,
8x=-8, x=-1
Substitution of x by its value,
2(-1)+4y=10
-2+4y=10
4y=12
y=3
The solutions to the system are, x=-1 and y=3

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Let call those integers x and y, if they are consecutives y=x+1.
x+(x+1)=x(x+1)-11 The sum of two consectutive positive integers is 11 less than their product.
2x+1=x^2+x-11
0=x^2-x-12
Using the general formula for solving quadratic equations we get x1=4,x2=-3
Then if the integers are positive they are 4 and 5, 4+5=9, 4*5=20, 20-9=11,
And if they are negative they are -3 and -2, -3+(-2)=-5, -3*(-2)=6, 6-(-5)=11.

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If you want a point in which the x is opposite to y the point is (x,-x),
To find such point we have to solve the equation for y=-x
3x-y = 12,
y=3x-12,
(-3)=3x-12,
-3+12=3x,
then 9=3x,
finally we get x=3
Therefore the point is (3,-3), In wich the x and the y are opposites.

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Sum:

and,
Product:

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Solving
15+20=2x+2x, 35=4x, x=35/4
6x+2x=20-4, 8x=16, x=2
5x-17=30x+3, -20=25x, x=-4/5
3(x+2-3x)=20+2(x-2), 3x+6-9x=20+2x-4, -10=8x, -5/4=x
10(x+5-3)=5x-3, 10x+50-30=5x-3, 5x=-23, x=-23/5
4x-10+30=4x+7x-8, 28=7x, 4=x
5x-10+2(3+x-2x)=8x, 5x-10+6+2x-4x=8x, -4=5x, -4/5=x
3x-2=4(-2+3x)+20, 3x-2=-8+12x+20, -14=9x, x=-14/9

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To eliminate the denominator you have to multiply all the expression by (x-1)(x-2)(2).
1(x-1)(x-2)(2)/(x-1)-1(x-1)(x-2)(2)/(x+2)=1(x-1)(x-2)(2)/2
Reducing common factors we get:
2(x-2)-2(x-1)=(x-1)(x-2)

Now we have to expand and reduce.

2x-4-2x+2=x^2-2x-x+2 Expanding
0=x^2-2x-x+2x-2x+4-2 Reducing
0=x^2-3x+2 Final equation
Know we have to solve this problem, you can use the general formula.

You will get as solutions x=1,x=2

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At first you must put the last number in the other side of the equation, so we get:
4x^2-48=0, 4x^2=48
Now we divide by 4 in both side of the equations and we get:
4x^/4=48/4, and then we get, x^2=12, now and to finish we have to get the square root of 12.

And now taking out the for of the square root we get
The solution of the equation.

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You can see the solution of the system when both equation have a common point in the graph. If they do not have a common point, I mean that if they are parallel, then the system do not have any solution. If both graph of the equations are the same then the system have infinite solution.
A system with one solution:

You can see the solutions in the graph, x=-1,and y=-3.
A system without any solution:

As you see in this system both lines are parallel, then they do not have any solution.
A system with infinite solutions:

In this system the graph of both equations is the same and therefore there are infinite solutions, for every point on the line.

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x^2-5x+6>2
Is the same as x^2-5x+4>0
Applying the general formula:

We get the solution x=1,x=4.
Therefore the solution is the union of two intervals minus the values for which the equation is equal to cero (-inf,1)U(4,+inf)-(1,4)

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(1) Here is the first
m^3+6m^2+9m=0 , This polynomial of degree 3 is divisible by m, therefore we divede by m and get:
m(m^2+6m+9)=0 Now we have to solve the polynomial of degree two, and the solution is m=-3, and is unique. Finally we get
m(m+3)(m+3) As the factorization of m^3 + 6m^2 + 9m
(2) Here is the second
b^2-6b-10=0, First have to solve the equation, find the roots, we can do this using the general formula or using more advance methods. Using the formula we get as a result b=3+sqrt(19), and b=3-sqrt(19). Then the factors will be
(b-(3-sqrt(19))(b-(3+(sqrt(19))

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Translating this to an equation you get:
1/10+1/2+d/50=70
Solving the equation we get:
5+25+d=70
d=70-30
d=40 It is the 3rd distance travelled

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-4<5+x<0 and x<0 Because it is negative
-4=5+x
-9=x Therefore x must be greater than -9
5+x=0
x=-5 Therefore x must be less than -5
The solution of the problem is the open interval of the numbers greater than -9 and less than -5, S=(-6,-7,-8) Are the threes negative integers that are solution to the problem.