In the following equation, what are possible values for g and h? (5 raised to the power of g times h) = 5 raised to the power of 12
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In the following equation, what are possible values for g and h?
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To solve the equation ( 5^{gh} = 5^{12} ), we can use the property of exponents that states if the bases are the same, the exponents must be equal. Therefore, we can set the exponents equal to each other:
[ gh = 12 ]
Now, to find possible values for ( g ) and ( h ), we can think of pairs of numbers that multiply to 12. Here are a few examples:
1. ( g = 1 ) and ( h = 12 ) (since ( 1 times 12 = 12 ))
2. ( g = 2 ) and ( h = 6 ) (since ( 2 times 6 = 12 ))
3. ( g = 3 ) and ( h = 4 ) (since ( 3 times 4 = 12 ))
4. ( g = 4 ) and ( h = 3 ) (since ( 4 times 3 = 12 ))
5. ( g = 6 ) and ( h = 2 ) (since ( 6 times 2 = 12 ))
6. ( g = 12 ) and ( h = 1 ) (since ( 12 times 1 = 12 ))
Thus, there are multiple possible pairs of values for ( g ) and ( h