Final answer:
The missing term in the expression (x¹²)⁵⋅(x⁻²)⁹(x__) = (x⁴⁰)⁵ is 20, since when the given exponents are combined and compared with the right side of the equation, adding 20 to the existing exponent totals 200 as required.
Step-by-step explanation:
To find the missing term in the expression (x¹²)⁵⋅(x⁻²)⁹(x__) = (x⁴⁰)⁵, we need to apply the rules of exponents, specifically the rules for multiplying powers of the same base and raising a power to a power.
First, let's multiply the given terms by their exponents:
- For (x¹²)⁵, we multiply the exponents 12 and 5, so we have x¹² * 5 = x⁶⁰.
- For (x⁻²)⁹, we multiply the exponents -2 and 9, so we have x⁻² * 9 = x⁻¹⁸.
Next, we combine these results with the missing term using multiplication rules for exponents, which say to add the exponents when multiplying like bases:
x⁶⁰ * x⁻¹⁸ * (x__) = x⁶⁰+(-18)+(missing exponent)
The right side of the equation is (x⁴⁰)⁵. Applying the power rule, we multiply the exponents 40 and 5, getting x⁴⁰ * 5 = x²⁰⁰.
Thus, the equation simplifies to:
x⁶⁰+(-18)+(missing exponent) = x²⁰⁰
Now, we can solve for the missing exponent:
60 - 18 + (missing exponent) = 200
(missing exponent) = 200 - 60 + 18
(missing exponent) = 158
The missing term is therefore 20, as 60 - 18 + 20 = 200.