randVar( ) randVar( ) randRangeNonZero( -5, 5 ) randRangeNonZero( -5, 5 ) randRangeNonZero( -5, 5 ) randRangeNonZero( -5, 5 ) randRangeNonZero( -5, 5 ) [ "^", [ "*", [ "^", BASE1, EXPDEN1 ], [ "^", BASE2, EXPDEN2 ] ], EXPDEN3 ] [ "*", [ "^", [ "^", BASE1, EXPDEN1 ], EXPDEN3 ], [ "^", [ "^", BASE2, EXPDEN2 ], EXPDEN3 ] ] [ "*", [ "^", BASE1, EXPDEN1 * EXPDEN3 ], [ "^", BASE2, EXPDEN2 * EXPDEN3 ] ]
randRangeNonZero( -5, 5 ) [ "^", [ "*", [ "^", BASE1, EXPNUM1 ], [ "^", BASE2, EXPNUM2 ] ], EXPNUM3 ] [ "*", [ "^", [ "^", BASE1, EXPNUM1 ], EXPNUM3 ], [ "^", [ "^", BASE2, EXPNUM2 ], EXPNUM3 ] ] [ "*", [ "^", BASE1, EXPNUM1 * EXPNUM3 ], [ "^", BASE2, EXPNUM2 * EXPNUM3 ] ] EXPNUM1 * EXPNUM3 - EXPDEN1 * EXPDEN3 EXPNUM2 * EXPNUM3 - EXPDEN2 * EXPDEN3 [ "*", [ "^", BASE1, EXP1 ], [ "^", BASE2, EXP2 ] ]

Simplify; express your answer in exponential form. Assume BASE1\neq 0, BASE2\neq 0.

\dfrac{\blue{expr( NUM )}}{\green{expr( DEN )}}

BASE1EXP1BASE2EXP2

enter a (possibly negative) integer for each exponent

To start, try simplifying the numerator and the denominator independently.

In the numerator, we can use the distributive property of exponents.

\blue{expr( NUM ) = expr( NUMHINT1 )}.

On the left, we have \blue{expr( [ "^", BASE1, EXPNUM1 ] )} to the exponent \blue{EXPNUM3}. Now \blue{EXPNUM1 \times EXPNUM3 = EXPNUM1 * EXPNUM3}, so \blue{expr( [ "^", [ "^", BASE1, EXPNUM1 ], EXPNUM3 ] ) = expr( [ "^", BASE1, EXPNUM1 * EXPNUM3 ] )}.

Apply the ideas above to simplify the equation.

\dfrac{\blue{expr( NUM )}}{\green{expr( DEN )}} = \dfrac{\blue{expr( NUMHINT2 )}}{\green{expr( DENHINT2 )}}.

Break up the equation by variable and simplify.

\dfrac{\blue{expr( NUMHINT2 )}}{\green{expr( DENHINT2 )}} = \dfrac{\blue{expr( [ "^", BASE1, EXPNUM1 * EXPNUM3 ] )}}{\green{expr( [ "^", BASE1, EXPDEN1 * EXPDEN3 ] )}} \cdot \dfrac{\blue{expr( [ "^", BASE2, EXPNUM2 * EXPNUM3 ] )}}{\green{expr( [ "^", BASE2, EXPDEN2 * EXPDEN3 ] )}} = BASE1^{\blue{EXPNUM1 * EXPNUM3} - \green{negParens( EXPDEN1 * EXPDEN3 )}} \cdot BASE2^{\blue{EXPNUM2 * EXPNUM3} - \green{negParens( EXPDEN2 * EXPDEN3 )}} = expr( ANS )

0 [ "^", [ "^", BASE1, EXPNUM1 ], EXPNUM3 ] [ "^", [ "^", BASE1, EXPNUM1 ], EXPNUM3 ] [ "^", BASE1, EXPNUM1 * EXPNUM3 ] EXPNUM1 * EXPNUM3 - EXPDEN1 * EXPDEN3 EXPNUM2 * EXPNUM3 - EXPDEN2 * EXPDEN3 [ "*", [ "^", BASE1, EXP1 ], [ "^", BASE2, EXP2 ] ]

Simplify; express your answer in exponential form. Assume BASE1\neq 0, BASE2\neq 0.

\dfrac{\blue{expr( NUM )}}{\green{expr( DEN )}}

BASE1EXP1BASE2EXP2

enter a (possibly negative) integer for each exponent

To start, try working on the numerator and the denominator independently.

In the numerator, we have \blue{expr( [ "^", BASE1, EXPNUM1 ] )} to the exponent \blue{EXPNUM3}. Now \blue{EXPNUM1 \times EXPNUM3 = EXPNUM1 * EXPNUM3}, so \blue{expr( NUM ) = expr( NUMHINT2 )}.

In the denominator, we can use the distributive property of exponents.

\green{expr( DEN ) = expr( DENHINT1 )}.

Simplify using the same method from the numerator and put the entire equation together.

\dfrac{\blue{expr( NUM )}}{\green{expr( DEN )}} = \dfrac{\blue{expr( NUMHINT2 )}}{\green{expr( DENHINT2 )}}.

Break up the equation by variable and simplify.

\dfrac{\blue{expr( NUMHINT2 )}}{\green{expr( DENHINT2 )}} = \dfrac{\blue{expr( [ "^", BASE1, EXPNUM1 * EXPNUM3 ] )}}{\green{expr( [ "^", BASE1, EXPDEN1 * EXPDEN3 ] )}} \cdot \dfrac{\blue{1}}{\green{expr( [ "^", BASE2, EXPDEN2 * EXPDEN3 ] )}} = BASE1^{\blue{EXPNUM1 * EXPNUM3} - \green{negParens( EXPDEN1 * EXPDEN3 )}} \cdot BASE2^{- \green{negParens( EXPDEN2 * EXPDEN3 )}} = expr( ANS ).