Why x by x is NOT equal to squared x within exponents?

Question

Normally you can write $x*x=x^2$

But if you are operating within exponents, $a^{x*x} \neq a^{x^{2}}$ as the latter is equal to $a^{2x}$.

Is it a problem of notation ?

[Edited]

Thank you to having helped me to solve this problem. It is indeed a problem of notation, in the sense that parenthesis matters. From wikipedia: "Without parentheses to modify the order of calculation, by convention the order is top-down, not bottom-up."

$$a^{x*x} = a^{x^{2}} = a^{(x^{2})} \neq (a^{x})^{2} = a^{x*2}$$

${(a^x)^2}$ is not the same as $a^{(x^2)}$. Additional and multiplication are associative, but exponentiation isn't. — MJD, Apr 24 '15 at 06:18

score 4 · Accepted Answer · answered Apr 24 '15 at 05:22

4

The expression $a^{x\cdot x}$ is equal to $a^{x^2}$ because $x\cdot x = x^2$.

However, $a^{x} a^{x} = a^{x+x} = a^{2x}$ because

$$\underset{x \text{ times} }{(a \cdots a )}\underset{x \text{ times} }{(a \cdots a)} = \underset{2x \text{ times} }{(a \cdots a)}. $$

answered Apr 24 '15 at 05:22

Andrey Kaipov

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That's only true if $x$ is an integer. The equation above it is true because of the laws of exponents. – marty cohen Apr 24 '15 at 05:44

Why x by x is NOT equal to squared x within exponents?

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