This exercise covers exponential arithmemtic with (mainly negative, when possible) rational bases and (mainly nonunit numerator) rational exponents. This exercise covers the last chunk of material from the Level 3 Exponents video.

twoBasesOneRoot() random() < .75 VALS.root isOdd( EXP_D ) && ( random() < .75 ) VALS.base_1 VALS.base_2 EXP_NEG ? BASE_D : BASE_N EXP_NEG ? BASE_N : BASE_D BASE_NEG round( pow( EXP_NEG ? BASE_D : BASE_N, 1 / EXP_D_ORIG ) ) round( pow( EXP_NEG ? BASE_N : BASE_D, 1 / EXP_D_ORIG ) ) function() { var max_exp = min( maxReasonableExp( ROOT_N_ORIG ), maxReasonableExp( ROOT_D_ORIG ) ); var exp_n; while ( exp_n === undefined || exp_n % EXP_D_ORIG === 0 ) { exp_n = randRange( 2, max_exp ); } return exp_n; }() getGCD( EXP_N_ORIG, EXP_D_ORIG ) EXP_N_ORIG / GCD EXP_D_ORIG / GCD round( pow( EXP_NEG ? BASE_D : BASE_N, 1 / EXP_D ) ) round( pow( EXP_NEG ? BASE_N : BASE_D, 1 / EXP_D ) ) ROOT_NEG && isOdd( EXP_N ) round( pow( ROOT_N, EXP_N ) ) round( pow( ROOT_D, EXP_N ) )

\Large fracParens( BASE_N, BASE_D )^{fracSmall( ( EXP_NEG ? -1 : 1 ) * EXP_N, EXP_D )}

SOL_N / SOL_D

= fracParens( BASEF_N, BASEF_D )^{fracSmall( EXP_N, EXP_D )}

= \left(fracParens( BASEF_N, BASEF_D )^{fracSmall( 1, EXP_D )}\right)^{EXP_N}

To simplify fracParens( BASEF_N, BASEF_D )^{fracSmall( 1, EXP_D )}, figure out what goes in the blank:
\left(? \right)^{abs( EXP_D )}=frac( BASEF_N, BASEF_D )

To simplify fracParens( BASEF_N, BASEF_D )^{fracSmall( 1, EXP_D )}, figure out what goes in the blank:
\left(\color{blue}{frac( ROOT_N, ROOT_D )}\right)^{abs( EXP_D )}=frac( BASEF_N, BASEF_D )

so \quadfracParens( BASEF_N, BASEF_D )^{fracSmall( 1, EXP_D )}=frac( ROOT_N, ROOT_D )

So fracParens( BASEF_N, BASEF_D )^{fracSmall( EXP_N, EXP_D )}=\left(fracParens( BASEF_N, BASEF_D )^{fracSmall( 1, EXP_D )}\right)^{EXP_N}=fracParens( ROOT_N, ROOT_D )^{EXP_N}

= fraction( ROOT_N, ROOT_D, true, true, false, true )^{EXP_N}

= v