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 Simplifying expressions with exponents
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{\color{orange}{expr( NUM )}}{\color{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.

\color{orange}{expr( NUM ) = expr( NUMHINT1 )}.

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

Apply the ideas above to simplify the equation.

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

Break up the equation by variable and simplify.

\dfrac{\color{orange}{expr( NUMHINT2 )}}{\color{green}{expr( DENHINT2 )}} = \dfrac{\color{orange}{expr( [ "^", BASE1, EXPNUM1 * EXPNUM3 ] )}}{\color{green}{expr( [ "^", BASE1, EXPDEN1 * EXPDEN3 ] )}} \cdot \dfrac{\color{orange}{expr( [ "^", BASE2, EXPNUM2 * EXPNUM3 ] )}}{\color{green}{expr( [ "^", BASE2, EXPDEN2 * EXPDEN3 ] )}} = BASE1^{\color{orange}{EXPNUM1 * EXPNUM3} - \color{green}{negParens( EXPDEN1 * EXPDEN3 )}} \cdot BASE2^{\color{orange}{EXPNUM2 * EXPNUM3} - \color{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{\color{orange}{expr( NUM )}}{\color{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 \color{orange}{expr( [ "^", BASE1, EXPNUM1 ] )} to the exponent \color{orange}{EXPNUM3}. Now \color{orange}{EXPNUM1 \times EXPNUM3 = EXPNUM1 * EXPNUM3}, so \color{orange}{expr( NUM ) = expr( NUMHINT2 )}.

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

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

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

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

Break up the equation by variable and simplify.

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