document.write( "Question 679089: How can I explain an irrational exponent? \n" ); document.write( "

Algebra.Com's Answer #421814 by MathLover1(20850) $\"\"$ $\"About$

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recall that, by definition, an exponent can be any number, real or complex, negative or positive, rational or $\"irrational\"$ , algebraic or transcendental.\r
\n" ); document.write( "\n" ); document.write( " $\"x%5E%28pi%29\"$ is an example of an irrational exponent\r
\n" ); document.write( "\n" ); document.write( "or $\"2%5Esqrt%282%29\"$ ....how to do this one?\r
\n" ); document.write( "\n" ); document.write( "we will do it by $\"approximating\"$ the $\"irrational\"$ exponent by $\"rational\"$ exponents that are closer and closer to the irrational one\r
\n" ); document.write( "\n" ); document.write( "In example $\"2%5Esqrt%282%29\"$ , you want the exponent $\"sqrt%282%29\"$ to be $\"approximately\"$ equal to $\"+1\"$ , so the first approximation to $\"2%5Esqrt%282%29\"$ is $\"2%5E1+=+2\"$ . \r
\n" ); document.write( "\n" ); document.write( "or, go one step further with $\"sqrt%282%29=1.4+=14%2F10+=7%2F5\"$ , so the second approximation to $\"2%5Esqrt%282%29\"$ is $\"2%5E%287%2F5%29=root%285%2C2%5E7%29\"$ which is approximately $\"2.639\"$ \r
\n" ); document.write( "\n" ); document.write( "or three significant figures,
\n" ); document.write( " $\"sqrt%282%29+=+1.41+=+141%2F100\"$ , so you have to take a $\"root%28100%2C2%5E141%29\"$ which is approximately $\"2.65737\"$ Continuing this way, we get the sequence of approximations:\r
\n" ); document.write( "\n" ); document.write( " $\"2\"$ , $\"2.639\"$ , $\"2.6574\"$ , $\"2.66475\"$ , $\"2.665119\"$ , $\"2.6651375\"$ , $\"2.66514310\"$ , ...\r
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