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Exponents And Powers

Question
CBSEENMA8002744
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Simplify :  space space fraction numerator 2 to the power of negative 5 end exponent cross times 10 to the power of negative 5 end exponent cross times 125 over denominator 5 to the power of negative 9 end exponent cross times 6 to the power of negative 5 end exponent end fraction

Solution

Since   125 space equals space 5 space cross times space 5 space cross times space 5 space equals space 5 cubed
           10 to the power of negative 5 end exponent space equals space left parenthesis 2 cross times 5 right parenthesis to the power of negative 5 end exponent space equals space 2 to the power of negative 5 end exponent space cross times space 5 to the power of negative 5 end exponent 6 to the power of negative 5 end exponent space equals space left parenthesis 2 cross times 3 right parenthesis to the power of negative 5 end exponent space equals space 2 to the power of negative 5 end exponent space cross times space 3 to the power of negative 5 end exponent

∴   fraction numerator 2 to the power of negative 5 end exponent cross times 10 to the power of negative 5 end exponent cross times 125 over denominator 5 to the power of negative 9 end exponent space cross times space 6 to the power of negative 5 end exponent end fraction space equals space fraction numerator 2 to the power of negative 5 end exponent cross times 2 to the power of negative 5 end exponent cross times 5 to the power of negative 5 end exponent cross times 5 cubed over denominator 5 to the power of negative 9 end exponent cross times 2 to the power of negative 5 end exponent cross times 3 to the power of negative 5 end exponent end fraction
                     equals open parentheses 2 to the power of negative 5 end exponent over 2 to the power of negative 5 end exponent close parentheses space cross times space fraction numerator 5 to the power of negative 5 plus left parenthesis negative 5 right parenthesis plus 3 end exponent over denominator 5 to the power of negative 9 end exponent cross times 2 to the power of negative 5 end exponent cross times 3 to the power of negative 5 end exponent end fraction
                     equals 1 space cross times space fraction numerator 5 to the power of negative 7 end exponent over denominator 5 to the power of negative 9 space end exponent cross times space 3 to the power of negative 5 end exponent end fraction
                      equals fraction numerator 1 cross times 5 to the power of 9 cross times 3 to the power of 5 over denominator 5 to the power of 7 end fraction
                      equals 1 cross times 5 to the power of 9 minus 7 end exponent cross times 3 to the power of 5
                       = 1 space cross times space 5 squared space cross times space 3 to the power of 5
                       = <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
                       = 25 cross times 243 = 6075.
                      

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