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How does one add these two terms with exponents in the numerator like $h^2+\frac{h^2}{4}$ ?

According to my lesson on Khan Academy, one should get $h^2\left(1 + \frac{1}{4}\right)$.

However, intuitively, it would seem that one would get $\frac{4h^2}{4}+\frac{h^2}{4}$ having first taken a common denominator and then $5\frac{h^2}{4}$.

After having searched for clarification, none of the search results really helped me to derive the answer. Hopefully this will not add, as such, a redundant post. Please clarify.

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Your answer and the Khan academy answer are the same.

Khan Academy: $$h^2\left(1+\frac14\right)$$ You: $$5\frac{h^2}4=\frac54h^2$$ But $$1+\frac14=\frac44+\frac14=\frac54$$So the Khan academy answer becomes $$h^2\cdot\frac54=\frac54h^2$$ which is the same as yours.

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note that $$\frac{h^2}{4}=\frac{1}{4}\cdot h^2$$ thus we have $$\frac{4}{4}\cdot h^2+\frac{1}{4}\cdot h^2=\frac{5}{4}\cdot h^2$$

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