Why does $(3^{1/2})(10^{1/2})=30^{1/2}$ but $(3a^2)(10a^2)=30a^4$?

Question

$(3a^2)(10a^2)=30a^4$? In that equation the exponents are added.

Why does $(3^{1/2})(10^{1/2})=30^{1/2}$. In that equation the exponents are not added.

Why?

Answer 1

You seem to be drawing a sort of false equivalency between multiplying $(a^2)(a^2)$ and multiplying $(3^{1/2})(10^{1/2})$. In the first case, the quantities being raised to the exponents are equal; thus, we may combine and get $a^4$. In the second case, the quantities are different; we may not, therefore, combine them and add the exponents. In terms of variables, we could look at $(a^2)(b^2)$ where $a \neq b$; in this case, we cannot combine, so we must content ourselves with writing $(ab)^2$. But if we have $(3^{1/2})(3^{1/2})$, we can add the exponents and find that the answer is 3.

Answer 2

First, notice the difference in both of the questions. The first question has a $\frac{1}{2}$ power or in other words, both the quantities are being square rooted and also, the bases are not the same. So, it's more like

$$(3^{1/2})(10^{1/2}) \implies \sqrt{3}\cdot \sqrt{10}$$

Now, according to some exponent rules if we something like $\sqrt{a}\cdot \sqrt{b}$ and both $a$ and $b$ are greater than zero then we can combine them like this $\sqrt{ab}$. So, for your question, we would have

$$\sqrt{3}\cdot \sqrt{10} = \sqrt{10\cdot 3} = \sqrt{30} \quad \text{OR} \quad 30^{\frac{1}{2}}$$

Now, for your second question. Notice that the exponents are on bases which are the same.

$$(3\color{red}{a}^\color{blue}{2})(10\color{red}{a}^\color{blue}{2})=30\color{red}{a}^\color{blue}{4}$$

Since, the bases are the same you can add the exponents but in the previous case the bases weren't the same and we had to apply a different rule to get something meaningful. So, that's reason why

$$(3^{1/2})(10^{1/2})=30^{1/2}$$ and $$(3a^2)(10a^2)=30a^4$$

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