smc

Write out the numbers using the definition of base numbers. $$554_b=5b^2+5b+4$$ $$24_b=2b+4$$ Since $554_b=(24_b{)}^2$, we can write an equation. $$5b^2+5b+4=(2b+4{)}^2$$ $$5b^2+5b+4=4b^2+16b+16$$ $$b^2-11b-12=0$$ $$(b-12)(b+1)=0$$ Since base numbers must be positive, $b$ in base 10 equals $\boxed{12}$.