How do you simplify $(8.39 \times 10^{- 7}) (4.53 \times 10^{9})$ $(8.39 xx 10^-7)(4.53 xx 10^9)$ ?

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How do you simplify $(8.39 \times 10^{- 7}) (4.53 \times 10^{9})$ $(8.39 xx 10^-7)(4.53 xx 10^9)$ ?

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Answer 1 · 2018-03-19T19:45:17

$(8.39 \times 10^{- 7}) \times (4.53 \times 10^{9})$ $(8.39 xx 10^-7)xx(4.53 xx10^9)$

$= 3.80067 \times 10^{3}$ $=3.80067 xx 10^3$

Explanation:

Consider the product of $3 x^{- 7} \times 7 x^{9}$ $" "3x^-7 xx 7x^9$

$\to$ $rarr$ multiply the numbers and add the indices of like bases.

$= 21 x^{- 7 + 9}$ $= 21 x^(-7+9)$

$= 21 x^{2}$ $=21x^2$

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

$\to$ $rarr$ multiply the numbers and add the indices of like bases.

$(8.39 \times 10^{- 7}) \times (4.53 \times 10^{9})$ $(8.39 xx 10^-7)xx(4.53 xx10^9)$

$38.0067 \times 10^{- 7 + 9}$ $38.0067 xx 10^(-7+9)$

$38 .0067 \times 10^{2} \leftarrow$ $color(blue)(38).0067 xx 10^2" "larr$ not in scientific notation.

$= 3.80067 \times 10^{3}$ $=3.80067 xx 10^3$

Answer 2 · 2018-03-19T19:46:24

See a solution process below:

Explanation:

First, rewrite the expression as:

$(8.39 \times 4.53) \times (10^{- 7} \times 10^{9}) \Rightarrow$ $(8.39 xx 4.53) xx (10^-7 xx 10^9) =>$

$38.0067 \times (10^{- 7} \times 10^{9})$ $38.0067 xx (10^-7 xx 10^9)$

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

$x^{a} \times x^{b} = x^{a + b}$ $x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) + color(blue)(b))$

$38.0067 \times (10^{- 7} \times 10^{9}) \Rightarrow$ $38.0067 xx (10^color(red)(-7) xx 10^color(blue)(9)) =>$

$38.0067 \times 10^{- 7 + 9} \Rightarrow$ $38.0067 xx 10^(color(red)(-7)+color(blue)(9)) =>$

$38.0067 \times 10^{2}$ $38.0067 xx 10^2$

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

$3.80067 \times 10^{2 + 1} \Rightarrow$ $3.80067 xx 10^(2+1) =>$

$3.80067 \times 10^{3}$ $3.80067 xx 10^3$

Or, in standard terms:

$3800.67$ $3800.67$

Answer 3 · 2018-03-19T19:52:40

By first combining the exponential terms, you can simplify the problem significantly, and get $3.80067 \times 10^{3}$ $3.80067xx10^3$ 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:

$(8.39 \times 10^{- 7}) (4.53 \times 10^{9}) = 8.39 \times 4.53 \times 10^{9} \times 10^{- 7}$ $(8.39xx10^(-7))(4.53xx10^9)=8.39xx4.53xx10^9xx10^(-7)$

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

$8.39 \cdot 4.53 = 38.0067 = 3.80067 \times 10^{1}$ $8.39*4.53=38.0067=3.80067xx10^1$

Re-writing the equation:

$3.80067 \times 10^{1} \times 10^{9} \times 1010^{- 7}$ $3.80067xx10^1xx10^9xx1010^(-7)$

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:

$x^{n} \times x^{z} = x^{n + z}$ $x^nxx x^z=x^(n+z)$

using that principle:

$10^{1} \times 10^{9} \times 10^{- 7} = 10^{1 + 9 - 7} = 10^{3}$ $10^1xx10^9xx10^(-7)=10^(1+9-7)=10^3$

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

$3.80067 \times 10^{3}$ $color(red)(3.80067xx10^3)$

Answer 4 · 2018-03-19T20:36:52

Suppose the idea of decimals is giving you a problem.

$3.80067 \times 10^{3}$ $3.80067xx10^3$

Explanation:

$Consider: 8.39$ $color(blue)("Consider: "8.39)$
this is the same as $839 \times \frac{1}{100} = 839 \times 10^{- 2}$ $839xx1/100 = 839xx10^(-2)$

So we have:

$[8.39] \times 10^{- 7} \to [839 \times 10^{- 2}] \times 10^{- 7} = 839 \times 10^{- 9}$ $[ 8.39 ]xx10^(-7) ->[839xx10^(-2)]xx10^(-7) = 839xx10^(-9)$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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

Using the same argument as above we have:

$[453 \times 10^{- 2}] \times 10^{9} = 453 \times 10^{7}$ $[453xx10^(-2)]xx10^9= 453xx10^7$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$Putting it all together$ $color(blue)("Putting it all together")$

$(839 \times 10^{- 9}) (453 \times 10^{7}) = 839 \times 453 \times \frac{10^{7}}{10^{9}}$ $(839xx10^(-9))(453xx10^7) =839xx453xx10^7/10^9$

$380067 \times \frac{1}{10^{2}}$ $380067xx1/10^2$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$380067 \times \frac{1}{10} \times \frac{1}{10} = 3800.67$ $380067xx1/10xx1/10 = 3800.67$

$3800.67 is the same as 380.067 \times 10$ $3800.67" is the same as "380.067xx10$
$3800.67 is the same as 38.0067 \times 10 \times 10$ $3800.67" is the same as "38.0067xx10xx10$
$3800.67 is the same as 3.80067 \times 10 \times 10 \times 10$ $3800.67" is the same as "3.80067xx10xx10xx10$

So in Scientific notation we have: $3.80067 \times 10^{3}$ $3.80067xx10^3$

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How do you simplify $(8.39 \times 10^{- 7}) (4.53 \times 10^{9})$ $(8.39 xx 10^-7)(4.53 xx 10^9)$ ?

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