Answer:
4.171×10⁶
Step-by-step explanation:
We can rewrite 97 as ...
97 = 0.97 × 100 = 0.97×10²
Then the product of the two numbers is ...
(97)(4.3×10⁴) = (0.97×10²)(4.3×10⁴) = (0.97×4.3)×(10²⁺⁴) = 4.171×10⁶
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More detailed explanation
You can always multiply and divide anything by the same number and the value does not change. Here we multiply 97 by 1 in the form of 100/100.
[tex]97=97\cdot\dfrac{100}{100}=\dfrac{97}{100}\cdot 100=0.97\cdot 100[/tex]
Of course, a power of ten can be written as such:
[tex]100=10^2\\\\0.97\cdot 100=0.97\cdot 10^2[/tex]
We use these facts to convert numbers to and from scientific notation.
Here, we want the result to have one digit to the left of the decimal point, so we don't want to multiply 4.3 by 9.7. That would give us 41.71, which has two digits left of the decimal point. Rather than adjust the power of 10 of the answer, we choose to adjust the power of ten of 97 so the product has one digit left of the decimal point. (It's called planning ahead. It saves some steps and some confusion.)
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The rules of exponents apply to powers of 10 just as they apply to variables or expressions:
(10^a)(10^b) = 10^(a+b)
And, of course, the commutative and associative properties* of multiplication apply to the product.
(0.97×10²)(4.3×10⁴) = 0.97×(10²×4.3)×10⁴ . . . . associative property
= 0.97×(4.3×10²)×10⁴ . . . . . . . . . . . . . . . . . . . . . commutative property
= (0.97×4.3)×(10²×10⁴) . . . . . . . . . . . . . . . . . . . . associative property
= (4.171)×(10²⁺⁴)
= 4.171×10⁶
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* The reason why you study these properties is so you can see the ways you can rearrange an expression to make it convenient to write or to evaluate. In any rearrangement you do, you want to make sure NOT to change the value.
