I know $(-7)^2=49$.
I know $\sqrt{x^2}=|x|$
But how come if you take the root of both sides of the first line it looks like this:
$\sqrt{(-7)^2}=\sqrt{49}\longrightarrow-7=7$?
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2If you know that $\sqrt{x^2}=|x|$, then you should automatically also know that $\sqrt{(-7)^2}=|-7|=7$, not $-7$... – Hans Lundmark Oct 12 '15 at 06:06
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$f(x)=x^2$ is not an injective $1-1$ function on the real numbers – Henry Oct 12 '15 at 06:06
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2Because, as you state, $\sqrt x= |x| \rightarrow \sqrt{(-7)^2}=|-7|=7$. – Lonidard Oct 12 '15 at 06:07
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The absolute value of both is equal. $\sqrt{(-7)^2}=\sqrt{49}=\text{POSITIVE}\boxed7$. But $-\sqrt{(-7)^2}=-\sqrt{49}=\text{NEGATIVE}\boxed7$. And |7|=|-7|=7$ – Aditya Agarwal Oct 12 '15 at 08:00
2 Answers
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You wrote:
$$\sqrt{(-7)^2}=\sqrt{49}\longrightarrow-7=7$$
But, you also wrote:
$$\sqrt{x^2}=|x|$$
So, how about:
$$\sqrt{(-7)^2}=\sqrt{49}\longrightarrow|-7|=7$$
MathAdam
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But the reason $\sqrt{x^2}=|x|$ is because there are two potential solutions, not one. This relationship isn't an axiom or something like that, it's a relationship that was noticed due to the potential number of solutions. Your answer doesn't make sense to me. Arithmetic doesn't always work the same way as algebra. You're telling me that: $((-7)^2)^{1/2}=7$? – Zduff Oct 13 '15 at 02:41
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1The absolute value is used because the codomain of the square root function is the non-negative reals. Negative solutions are not allowed. I did not choose to represent it this way. You did. I used the format you provided and substituted the digits. I'm sorry this doesn't make sense to you. :( – MathAdam Oct 13 '15 at 03:00
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@zduff Yes, I'm telling you that $\sqrt {(-7)^2}=7$. That's how it's defined. Moreover, that's what you wrote when you wrote that $sqrt{x^2}=|x|. You said it before I did. – MathAdam Oct 13 '15 at 03:12
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The part I don't understand is how: $\sqrt{(-7)^2}=((-7)^2)^{1/2}\neq((-7)^{1/2})^2$ – Zduff Oct 13 '15 at 11:30
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I didn't choose to represent it that way. That's the way square root function is defined. – Zduff Oct 13 '15 at 12:09
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Never mind, found an answer that is satisfying. I realized that my conceptual problem was with the way square root functions are defined. They are defined to have a positive codomain so that they can remain a function. Here's a link: Proving square root of a square is the same as absolute value
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2When you wrotr, "I know $\sqrt{x^2}=|x|$ ," I thought you were satisfied with that definition. Sounds like it's the definition that was creating the problem for you. Glad you found your answer. – MathAdam Oct 13 '15 at 13:14