$(3a^2)(10a^2)=30a^4$? 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.


  • $\begingroup$ $(3a^2)(10a^2) = 30 a^4$, not $30^4$. The exponent is attached only to $a$. $\endgroup$ – user61527 Jul 15 '14 at 1:51
  • 3
    $\begingroup$ See if this is useful. $\endgroup$ – David Jul 15 '14 at 1:51
  • $\begingroup$ @T.Bongers Thanks, that's what I meant to write. I'll edit that. $\endgroup$ – Hal Jul 15 '14 at 1:51
  • $\begingroup$ If it were $(3a)^2(10a)^2$, the calculations would be analogical. $\endgroup$ – André Nicolas Jul 15 '14 at 1:51
  • $\begingroup$ @David it was, thank you. $\endgroup$ – Hal Jul 15 '14 at 12:16

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.


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.


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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