Compare the two Quantities!

Algebra Level 3

Which quantity is greater and by approximately how many times?

A = 5 10 A=5^{10} or B = 1 0 5 B=10^5

A and by 1000 A is equal to B B and by 100 B and by 1000 A and by 100

Hana Wehbi by Hana Wehbi , 51, USA

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3 solutions

Steven Adler
Jan 21, 2021

B: 10^5 = (2x5)^5 = 2^5 x 5^5

A: 5^10 = 5^5 x 5^5

Thus A/B = 5^5 / 2^5 = (5/2)^5 = 2.5^5

Every astronomer knows 2.5^5 ~ 100 because 5 units of magnitude are defined as a ratio of brightness of exactly 100, or 1 unit of magnitude is the 5th root of 100, almost exactly 2.5.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Thank you for sharing your solution.

Hana Wehbi - 4 months, 3 weeks ago
Chew-Seong Cheong
Nov 13, 2019

A B = 5 10 1 0 5 = 5 5 × 5 5 2 5 × 5 5 = ( 5 2 ) 5 = ( 10 4 ) 5 = 1 0 5 2 10 = 1 0 5 1024 ≈ 1 0 5 1 0 3 = 100 \dfrac AB = \dfrac {5^{10}}{10^5} = \dfrac {5^5\times 5^5}{2^5 \times 5^5} = \left(\dfrac 52\right)^5 = \left(\dfrac {10}4\right)^5 = \dfrac {10^5}{2^{10}} = \dfrac {10^5}{1024} \approx \dfrac {10^5}{10^3} = \boxed{100} .

Therefore A is greater and by about 100 times .

1 Helpful 0 Interesting 0 Brilliant 0 Confused

This solution makes it clear, thank you.

Hana Wehbi - 1 year, 7 months ago
Hana Wehbi
Nov 13, 2019

5 10 1 0 5 = 5 5 × 5 5 2 5 × 5 5 = 97.656 ≈ 100 \frac{5^{10}}{10^5} = \frac{5^5\times 5^5}{2^5\times 5^5} =97.656 \approx 100

0 Helpful 0 Interesting 0 Brilliant 0 Confused

You question should by "... and by approximately how many times ?" and answer should be "A and by about 100 times".

Chew-Seong Cheong - 1 year, 7 months ago

Thank you Sir.

Hana Wehbi - 1 year, 7 months ago

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