Asked by Keen
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Part 1
What is the value of n? Use pencil and paper. Explain why the exponent on the left side of the equation is not equal to the sum of the exponents on the right side.
4.5 times 10 Superscript 15equalsleft parenthesis 5 times 10 Superscript 10 right parenthesis left parenthesis 9 times 10 Superscript n right parenthesis
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Part 1
The value of n is
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GPT-5 mini
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n = 4
Work:
(5 × 10^10)(9 × 10^n) = (5·9) × 10^{10+n} = 45 × 10^{10+n}.
Write 45 as 4.5 × 10^1 to put the product in proper scientific notation:
45 × 10^{10+n} = (4.5 × 10^1) × 10^{10+n} = 4.5 × 10^{11+n}.
Now equate to the left side 4.5 × 10^{15}: 4.5 × 10^{15} = 4.5 × 10^{11+n}, so 15 = 11 + n, hence n = 4.
Why the exponent on the left is not simply 10 + n:
Because the coefficients 5 and 9 multiply to 45, which contributes an extra factor of 10 (45 = 4.5 × 10^1). That extra factor changes the overall power of 10, so you must normalize the coefficient to between 1 and 10 before equating exponents.
Work:
(5 × 10^10)(9 × 10^n) = (5·9) × 10^{10+n} = 45 × 10^{10+n}.
Write 45 as 4.5 × 10^1 to put the product in proper scientific notation:
45 × 10^{10+n} = (4.5 × 10^1) × 10^{10+n} = 4.5 × 10^{11+n}.
Now equate to the left side 4.5 × 10^{15}: 4.5 × 10^{15} = 4.5 × 10^{11+n}, so 15 = 11 + n, hence n = 4.
Why the exponent on the left is not simply 10 + n:
Because the coefficients 5 and 9 multiply to 45, which contributes an extra factor of 10 (45 = 4.5 × 10^1). That extra factor changes the overall power of 10, so you must normalize the coefficient to between 1 and 10 before equating exponents.
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