Question 25

If $$2^{3a+2}=4^{b+7}$$ and $$3^{a+10}=27^{2b+10}$$, then the value of $$a^2+b^2$$ is ______.

Correct Answer: 13

Solution

Comparing powers of $$2$$ gives $$3a+2=2b+14$$, so $$3a-2b=12$$. Comparing powers of $$3$$ gives $$a+10=6b+30$$, so $$a-6b=20$$. Solving these equations gives $$a=2$$ and $$b=-3$$, hence $$a^2+b^2=4+9=13$$.

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