Hello,
Just divide the LHS by 2 and the equation becomes :
5^21 * 2^21 = 10 ^ n
=> 10^21 = 10^n
n = 21.
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When we multiply two powers with the same base, we add the exponents.Jeffers wrote:Can someone explain this to me? I thought when you multiply powers you add them together?
However, when we multiply two bases with the same power, we multiply the bases and the power stays the same.
So:
2^21 * 5^21 = (2*5)^21 = 10^21
To help understand this, we can visualize 2^21 as a string of 21 2s, multiplied together. Similarly, we can visualize 5^21 as a string of 21 5s, multiplied together.
So, 2^21 * 5^21 is really just 21 2s and then 21 5s, all multiplied together.
Instead of writing all the 2s in a row and all the 5s in a row, we could have alternated them (since for multiplication, the order of terms is irrelevant).
So, we can visualize a string of 2*5*2*5*2*5... that contains 21 2s and 21 5s in total.
Now, let's put brackets around each pair of terms to get:
(2*5)*(2*5)*(2*5)... and we'll have 21 bracketed pairs.
Since 2*5 = 10 we can rewrite this as:
(10)*(10)*(10).... with 21 10s in a row, which of course we can rewrite as:
10^21.
Hope that helps!
Stuart Kovinsky | Kaplan GMAT Faculty | Toronto
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