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Question

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


A
2
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B
1
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C
3
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D
4
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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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