Question

# If $${ 4 }^{ a }+{ 4 }^{ a+1 }={ 4 }^{ a+2 }-176$$, what is the value of $$a$$?

A
2
B
1
C
3
D
4

Solution

## The correct option is C $$2$$The key to this problem is to express all of the exponential terms of the greatest common factor of the terms: $${4}^{a}$$. Using the addition rule (or the corresponding numerical examples), we get$${ 4 }^{ a }+{ 4 }^{ a+1 }={ 4 }^{ a+2 }-176$$$$176={4}^{a+2}-{4}^{a}-{4}^{a+1}$$$$176={4}^{a}\times ({4}^{2})-{4}^{a}-{4}^{a}.({4}^{1})$$$$176={4}^{a}\times ({4}^{2}-{4}^{0}-{4}^{1})$$$$176={4}^{a}\times (16-1-4)$$$$176={4}^{a}\times (11)$$$${4}^{a}=176\div 11=16$$$$a=2$$Mathematics

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