How do you simplify #(8.39 xx 10^-7)(4.53 xx 10^9)#?

4 Answers
EZ as pi
Mar 19, 2018

#(8.39 xx 10^-7)xx(4.53 xx10^9)#

#=3.80067 xx 10^3#

Explanation:

Consider the product of #" "3x^-7 xx 7x^9#

#rarr # multiply the numbers and add the indices of like bases.

#= 21 x^(-7+9)#

#=21x^2#

Exactly the same happens when you are multiplying numbers in scientific notation.

#rarr # multiply the numbers and add the indices of like bases.

#(8.39 xx 10^-7)xx(4.53 xx10^9)#

#38.0067 xx 10^(-7+9)#

#color(blue)(38).0067 xx 10^2" "larr# not in scientific notation.

#=3.80067 xx 10^3#

smendyka
Mar 19, 2018

See a solution process below:

Explanation:

First, rewrite the expression as:

#(8.39 xx 4.53) xx (10^-7 xx 10^9) =>#

#38.0067 xx (10^-7 xx 10^9)#

Now, we can use this rule for exponents to simplify the 10s terms:

#x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) + color(blue)(b))#

#38.0067 xx (10^color(red)(-7) xx 10^color(blue)(9)) =>#

#38.0067 xx 10^(color(red)(-7)+color(blue)(9)) =>#

#38.0067 xx 10^2#

To write this in scientific notation we need to move the decimal point one place to the left so we need to add #1# to the 10s exponent:

#3.80067 xx 10^(2+1) =>#

#3.80067 xx 10^3#

Or, in standard terms:

#3800.67#

Ben
Mar 19, 2018

By first combining the exponential terms, you can simplify the problem significantly, and get #3.80067xx10^3# as the result.

Explanation:

The first thing to do is rewrite the function so all of the multiplication is outside of parentheses, and the exponents are grouped together:

#(8.39xx10^(-7))(4.53xx10^9)=8.39xx4.53xx10^9xx10^(-7)#

Next, lets multiply the constants together, and use scientific notation to describe the result:

#8.39*4.53=38.0067=3.80067xx10^1#

Re-writing the equation:

#3.80067xx10^1xx10^9xx1010^(-7)#

Finally, we combine the notation terms. When you multiply two exponential numbers with the same base, it is the same as adding the exponents together, like so:

#x^nxx x^z=x^(n+z)#

using that principle:

#10^1xx10^9xx10^(-7)=10^(1+9-7)=10^3#

Now, lets re-write the equation one last time with all of our simplification:

#color(red)(3.80067xx10^3)#

Tony B
Mar 19, 2018

Suppose the idea of decimals is giving you a problem.

#3.80067xx10^3#

Explanation:

#color(blue)("Consider: "8.39)#
this is the same as #839xx1/100 = 839xx10^(-2)#

So we have:

#[ 8.39 ]xx10^(-7) ->[839xx10^(-2)]xx10^(-7) = 839xx10^(-9)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(blue)("Consider: "4.53)#

Using the same argument as above we have:

#[453xx10^(-2)]xx10^9= 453xx10^7#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Putting it all together")#

#(839xx10^(-9))(453xx10^7) =839xx453xx10^7/10^9#

#380067xx1/10^2#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#380067xx1/10xx1/10 = 3800.67#

#3800.67" is the same as "380.067xx10#
#3800.67" is the same as "38.0067xx10xx10#
#3800.67" is the same as "3.80067xx10xx10xx10#

So in Scientific notation we have: #3.80067xx10^3#

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