document.write( "Question 838174: Please could someone help with the following with workings out please: \r
\n" ); document.write( "\n" ); document.write( "16^2-n = (1/4)^n+1. \r
\n" ); document.write( "\n" ); document.write( "sixteen to the power two negative n equals a quarter to the power of n plus one.\r
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\n" ); document.write( "\n" ); document.write( "In my assignment I can work out the ones like 128^n = 8 and (1/4)^n = 8, but I'm getting stuck with the more complex ones. Please could you explain how you get to each stage if possible. Many thanks in advance, David. \r
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Algebra.Com's Answer #504890 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
First of all, please put exponents which are not just a variable or single number in parentheses. If you have not also explained your expression in English I would not have been sure what your equation was.

\n" ); document.write( " $\"16%5E%282-n%29+=+%281%2F4%29%5E%28n%2B1%29\"$
\n" ); document.write( "Equations with variables in the exponents can usually be solved using logarithms. But if it is possible to rewrite the equation so that each side is a power of the same base, then there is an easier and more accurate way than using logarithms. 16 and 1/4 are powers of each other but what powers these are may not be obvious. But they are both powers of 2 (which should be more obvious). So the solution will be easier if we start by rewriting each side as powers of 2:
\n" ); document.write( " $\"%282%5E4%29%5E%282-n%29+=+%282%5E%28-2%29%29%5E%28n%2B1%29\"$
\n" ); document.write( "For powers of a power the rule is multiply the exponents:
\n" ); document.write( " $\"2%5E%288-4n%29+=+2%5E%28-2n-2%29\"$

\n" ); document.write( "The next step is based on simple logic. The only way two powers of 2 can be equal, as the now equation says they are, is if the exponents are equal, too. So:
\n" ); document.write( "8-4n = -2n-2
\n" ); document.write( "Now we solve this simple equation. Adding 4n to each side:
\n" ); document.write( "8 = 2n-2
\n" ); document.write( "Adding 2:
\n" ); document.write( "10 = 2n
\n" ); document.write( "Dividing by 2:
\n" ); document.write( "5 = n

\n" ); document.write( "P.S. If we had used logarithms we might not have gotten an answer of 5. Because of the rounded decimals which a calculator uses, we might have gotten something like 5.00001 or 4.99998. So the method we used above should be used whenever possible. \n" ); document.write( "

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