The distance from earth to the moon is 2.4 × 10^5 miles. the distance from earth to the sun is 9.3 × 10^7 miles. the distance from earth to the sun is ? × 10^? miles greater than the distance from earth to the moon.

The answer is 9.276 * 10^7 miles

I'm using the asterisk symbol * instead of the x symbol to mean "multiply"

Explanation:

They want you to find the difference in those distances. This means you'll subtract the values (large minus small). Before you can subtract, you need to make sure that both exponents over the 10 are the same; otherwise, subtraction won't make much sense and you'll most likely get a bad answer.  Thankfully we can do a conversion.

Start with the larger exponent 10^7. Note how we can break it up like this:

10^7 = 10^(2+5) = 10^2*10^5 = 100*10^5

I used the rule a^(b+c) = a^b*a^c and then I simplified the value of 10^2 to be 100

The key is that we have a 10^5 in there.

So we can write 9.3 * 10^7 like so:

9.3 * 10^7 = 9.3 * 100*10^5

9.3 * 10^7 = (9.3 * 100)*10^5

9.3 * 10^7 = 930 * 10^5

The exponent portion 10^5 matches up with the other number's exponent portion

So we want to find the difference between the distances 2.4*10^5 and 930*10^5

Subtract the distances

(Larger) - (Smaller) = (930*10^5) - (2.4*10^5) = (930-2.4)*10^5 = 927.6*10^5

The exponent part 10^5 stays the same. The bases 930 and 2.4 subtract like any other pair of numbers.

The last step is to convert to scientific notation

927.6*10^5 = (927.6)*10^5

927.6*10^5 = (9.276*10^2)*10^5

927.6*10^5 = 9.276*(10^2*10^5)

927.6*10^5 = 9.276*10^(2+5)

927.6*10^5 = 9.276*10^7

which is the final answer.

I would avoid rounding the 9.276 to 9.3 because it would cause a bit of confusion. It would make the difference seem exactly the same as the distance from earth to the sun, but that's not the case here.

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!