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7 times the sum of M and 6 is expressed by: 7(M+6) and if you expand this, it becomes: 7M+42

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The surface area of a sphere with diameter D is given by:

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Since there are 4 qts in 1 gallon, you would divide the number of quarts (q) by 4 to find the number of gallons, so...

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Ok, you got a good start!
Let b = number of boys yesterday and (b+4) = the number of boys today and g = number of girls (This didn't change).
Yesterday: g = 5b
Today: g = 3(b+4)
Since g = g, then;
5b = 3(b+4) Apply the distributive property on the right side.
5b = 3b+12 Subtract 3b from both sides.
2b = 12 Divide both sides by 2.
b = 6
So, yesterday, there were 6 boys plus 5*(6) girls for a total of 36 children.
Today, there are 6+4 = 10 boys and 3*10 = 30 girls for a total 40 children.

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You are ok up to this point.
which simplifies to Now you multiply this by
Combining like-terms, you get:
So...

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G'day Sally;
Let x = the amount invested at 8% and the remainder ($9000-x) is invested at 9%
The amount of interest earned on these amounts can be expressed as:
x(0.08) This is amount earned at 8%.
($9000-x)(0.09) This is the amount earned at 9%
The sum (+) of these two amounts is given as $770.00, so you can write the equation to solve for x, the amount invested at 8%
(0.08)x + (0.09)($9000-x) = $770 Simplify and solve for x.
0.08x + $810 - 0.09x = $770 Subtract $810 from both sides.
0.08x-0.09x = -$40 Combine the x's on the left side.
-0.01x = -$40 Finally, divide both sides by -0.01 to get x by itself.
x = $4000
So $4000 was invested at 8% and $9000-$4000 = $5000 was invested at 9%
Let's check the solution:
0.08($4000) + 0.09($5000) = $320 + $450 = $770 The total interest earned.

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Solve for x:
Apply the power rule to the first term on the left side:
Now apply the quotient rule to the left side:
Rewrite this in its exponential form: --->
Multiply both sides by (1-x)
Expand the left side.
Add to both sides.
Factor the x on the right side.
Divide both sides by
Ths is the exact answer. If you want an approximate answer, then substitute to get:

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What is the answer to: ?
The answer depends entirely upon the value of x!
Did you forget to include that?

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Simplify:
====

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Divide:
divided byDo the first fraction by factoring the numerator and the denominator:
Cancel the indicated factors to leave you with:
Now divide by the second fraction, but remember to change to multiplication and invert the second fraction.
Cancel the indicated factors to leave you with:
Finally, reduce this fraction to its lowest terms.

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You are not alone in being mystified by logarithms!
To understand logarithms, you should be familiar with exponents because they are closely related.
Remember exponents, also called powers?
Here, the base is 10 and the exponent (power) is 2.
Here, the base is 2 and the exponent is 3.
So, here's a definition of logarithms:
" The logarithm of a number is the power to which the base of the logarithm must be raised to equal the number"
So a logarithm has a "base".
Commonly, this base is 10, but it can be any number. When the logarithm has a base of 10, it is called a "common logarithm" or "common log" for short.
Let's see how this works:
Let's suppose that we want to find the common log (base 10) of 100.
We would write:
You could show the base if you wanted to make absolutely sure that your reader undertood which base you were using by:
but, if no base is shown, and the notation is log or Log, then common logarithms are assumed.
What the above equation says is..."x is the power to which the base (10) must be raised to equal 100".
Well, you know that already because so x must equal 2, or

In fact, you can express the logarithm above in exponential a general form:
The base (b) is 10 and x = 2, so...

In more general terms, let b denote the base, we can write:
You would read this as: "The log, to the base b, of x = y" and this can be expressed in exponential form as:

Let's try a few more numbers, but we will stay with common (base 10) logs for now.
why, because
why, because
Now if you wanted to find the common log of a number that is not a multiple of the base (10), you would then have to resort to a calculator or a table of common logarithms.
For example:
approximately. The result is an irrational number (remember those?)
Most scientific or math calculators will have a logarithm key.
This is just an overview but you could log on to any good information site and type in Logarithms and you'd probably get more information on the topic than you could use.

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Think about this!
Through any three non-collinear points, there is exactly one plane!
The planes that you identify here must include the point E, so you can list all of the possible combinations of the points, A, B, C, D, with the point E.
E,A,B
E,A,C
E,A,D
E,B,C
E,B,D
E,C,D
So it seems that there are 6 distinct planes determined by the five points if point E is to be included in each one.

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You should include the zero for completeness, and notice that you start at an exponent of 9:
1,203,487,386 =
1,000,000,000 =
200,000,000 =
-------------------------
3,000,000 =
400,000 =
80,000 =
7,000 =
300 =
80 =
6 = Recall that

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Unfortunately, I don't have the math book from which your problem is taken.

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Presumably you would like to simplify this fraction:
Factor the numerator and the denominator.

Cancel the indicated factors.

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You have used the slope formula correctly: and substiuting the given points, you got:

Now you should reduce this fraction to its lowest terms:

Now let's look at the fractions.

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To answer your questions...NO! and NO! Let's look at the problem:
Simplify:

Let's do the numerator first: Factor the trinomial:
= Your work in this step was ok!
Now the denominator: Factor:
= You correct here as well!
Now we'll put 'em together:
At this point, can you see the cancellation possibilities?
=

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Some times, with mixture problems such as this one, it might be a little easier if you think in terms of just the antifreeze rather than the mixture.
You are told that Samantha needs 20 quarts of 50% antifreeze. First, change the percentage to its decimal equavalent (50% = 50/100 = 0.5).
Now 50% of 20 quarts of antifreeze mixture is really 0.5 (20) = 10 quarts of pure antifreeze.
The amount of 40% antifreeze mixture to be added is (20-x) quarts, that is this is the amount, added to the x quarts of pure antifreeze to get the 20 quarts.
Now let x = the amount of pure (100% = 1) antifreeze to be added to (20-x) quarts of 40% antifreeze mixture.
The amount of 40% antifreeze mixture is 0.4(20-x) quarts of antifreeze.
Now you can write an equation to solve for x, the number of quarts of pure antifreeze that Samantha needs.
Perform the indicated multiplication.
Combine the x's
Subtract 8 from both sides.
Finally, didvide both sides by 0.6 to get x by itself.
or x = 3 1/3
So Samantha needs to mix 3 1/3 quarts of pure antifreeze with 20-x = 20-3 1/3 = 16 2/3 quarts of 40% antifreeze mixture to obtain 20 quarts of 50% antifreeze mixture..

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All of your answers look good except for the first problem, part e.
Given: ...You said "This is not a polynomial" Why is it not?
It certainly qualifies as a polynomial (a trinomial in this case) because there are no terms with variables in the denominator and no terms with variables under a radical sign.
This polynomial is of degree 4 because of the

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I don't see the figure you allude to but it's easy enough imagine what it looks like.
In your semicircle, consider the radius from the center to the point of tangency (3, 4). This radius has a slope of (rise = 4 over run = 3). Now any line that is perpendicular to this radius. eg, the tangent line at (3, 4), will have a slope that is the negative reciprocal of the slope of this radius. So the slope of the tangent line will be
So now you can write:
But of course, you still need to find the value of b, the y-intercept, so substitute the x- and y-coordinate values of the point of tangency (3, 4) into this equation and solve for b.
Substitute x = 3 and y = 4
Solve for b.


So your final equation would be:
or, if you multiply through by 4 to clear the fractions, it becomes:

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Or did you mean "Scientific notation"?

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An equation for the line that is parallel to the line: y = -3x and which passes through the point (3, 0).
First, you should recall that parallel lines have the same slope.
The given line has a slope: m = -3, so the new line will have a slope of: m = -3.
So you can start by writing:
y = -3x+b Now you need to find the value of b, the y-intercept. This can be done by substituting the x- and y-coordinate values of the given point (3, 0) into the equation y = -3x+b and solving for b.
0 = -3(3) + b
0 = -9 + b
b = 9
Now you can write the final equation of the line that is parallel to the line whose equation is y = -3x and which passes through the point (3, 0)

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It's not clear how this problem is to be read!
1) or
2) I suspect that this is what you wanted, so let's go with it.
Substitute
Factor out .
Divide both sides by
Take the natural log of both sides.
Substitute:



Approximately.

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Let's use a modified FOIL (First-Outer-Inner-Last) method to multiply"
Now simplify this - remembering that when you multiply factors with the same base (x in this case) you will add the exponents of the factors.
Now combine like-terms.
...and this could be simplified to:

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You could try this:
Let x = the left page number and x+1 = the next consecutive page number.
You can write:
The product (this is a result of multiplying) of the consecutive page numbers is 9506
Subtract 9506 from both sides.

The roots are:
x = 97
x = -98
So from this, the page numbers are 97 and 98
The problem I have with this answer is that in most books, their left page numbers are even numbers, not odd numbers.

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You start by carefully folding down one corner of the 8.5 by 11 sheet of paper so that the top of the sheet is accurately alligned with the side of the sheet.
Crease the paper at the fold so that it maintains its shape (a triangle).
Now you fold up the bottom of the sheet (this has the shape of a rectangle with dimensions 8.5 by 2.5. The 2.5 comes from the 11.5-inch side minus the 8-inch side that was folded down in the first step.
Now you have a side of the twice-folded sheet that measures 6 inches, if you have folded very carefully and if your sheet was really 8.5 by 11 inches.

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Perhaps that is because the trinomial is not factorable...in other words, it's prime.
Let's rearrange the trinomial so that it's in standard form:

The factors, if there were any, would take the form:
and you need to have m*n = -20 and 3n+m = -3
The factors of -20 are:
-20 = -1*20 or 1*(-20) and 3(-1)+20 = 17, 3(1)+(-20) = -17
-20 = -2*10 or 2*(-10) and 3(-2)+10 = 4, 3(2)+(-10) = -4
-20 = -4*5 or 4*(-5) and 3(-4)+5 = -7, 3(4)+(-5) = 7
So, as you can see, there is no way to get a -3 from combining the factors of -20.
The trinomial is prime!

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Solve:

Subtract the second equation from the first equation to eliminate the y.
Divide both sides by 4.
Now substitute this value of x into either of the two given equations to solve for y. The second equation will work nicely.
Simplify.
Subtract 1 from both sides.
Multiply both sides by the multiplicative inverse of , that's


The solution is: (1/2, 9)

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Solve for x:
Multiply both sides by x
Now divide both sides by 147.12
Perform the indicated division on the right side.
Approximately.

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Solve for x:
Raise both sides to the power of
To the left side, apply the "product rule for exponents"

This is supposed to be: over and
Multiply top and bottom by
Now subtract 2 from both sides.

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Find: if:
Simply replace the x's with and evaluate.


Reduce
Simplify.