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What is the product of 8 times 10 -7th power and 5 times 10 4th power?

User JHeni
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Explanation:

if I understand you correctly, then the problem is

8×(10^-7) × 5×(10^4)

remember, like in additions, in pure multiplications we can permutate (mix) the factors in any way we want.

this is because of the associative property of "+" and "×" operations.

(a×b)×c = a×(b×c)

so, to be sure we know what we are doing, we group the factors based on their type (a power of 10 or not a power of 10) :

8×5 × (10^-7)×(10^4) = 40 × (10^-7)×(10^4)

to calculate the multiplication of the powers of 10 remember, even the base of factors is the same, then we simply add their exponents.

x^a × x^b = x^(a+b)

so, we have

40 × 10^(-7 + 4) = 40 × 10^-3

that is already a valid solution of the multiplication.

now (it was not required in the problem text, but I have to assume it might be in the context), remember the scientific notation :

any number can be represented in the form

a × 10^b

where "a" is a real number with only one digit (not 0) left of the decimal point. and at least one digit (even if it is 0) right of the decimal point.

e.g.

7.0837 × 10⁸

1.500086 × 10^-9

8.0 × 10²

so, what do we have to do to convert

40 × 10^-3

into a true scientific notation ?

well, 40 needs to turn into 4.0. so it has to be divided by 10 (multiplied by 10^-1).

and therefore, to keep the overall value of the number the same, we need to multiply the right side (the power of 10 factor) by 10 (= 10¹).

so,

40×(10^-1) × (10^-3)×(10^1) = 4.0 × 10^-2

that is our true solution.

User Bryan Schoen
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