What is the domain of x 2 3?

Cracking the Code: What’s the Real Story with x^(2/3)?

So, you’re wondering about the domain of x^(2/3), huh? It sounds like a complicated math problem, but trust me, we can break it down. Basically, the domain is just a fancy way of asking: “What numbers can I plug in for ‘x’ without the whole thing blowing up?”

Now, your first instinct might be to think, “Oh, it’s like a square root, so only positive numbers allowed!” And that’s a reasonable guess. But here’s where it gets interesting. This isn’t just a root; it’s a fractional exponent, and that opens up some possibilities.

Think of x^(2/3) in a couple of different ways. You can either:

See? Two different ways to get to the same answer.

Why Negatives Get a Pass

Here’s the kicker: you can take the cube root of any number, positive or negative. Remember that the cube root of -8 is -2. No problem! So, if you cube root first, like in our first option, you’re golden. You can plug in any number you want. Then, when you square it, you’ll always get a positive result.

What about squaring first? Well, when you square a number, the result is always positive or zero. And since you can take the cube root of any positive number (or zero), that works too!

So, surprisingly, the domain of x^(2/3) isn’t just positive numbers. It’s all real numbers!

A Word of Warning: Computers Can Be Sneaky

Now, before you go off thinking you’ve mastered fractional exponents, there’s a little wrinkle. Sometimes, computers (and even fancy calculators) don’t always play nice with these things.

I’ve seen it happen where software like Wolfram Alpha, when faced with x^(2/3), might give you a complex number answer when you’re expecting a real number. It’s defaulting to something called the “principal root,” which is a whole other can of worms. If that happens, try rewriting it as (cbrt(x))^2. That usually sets things straight.

Also, some graphing programs might be a bit cautious and only show the positive side of the graph. It’s like they’re playing it safe, even though the negative values are perfectly valid. It’s all down to how they approximate the function.

The Bottom Line

So, what’s the takeaway? The domain of f(x) = x^(2/3) is all real numbers. Don’t let anyone tell you otherwise! Just be aware that some software might try to trick you with complex numbers or incomplete graphs. Understanding the math behind it helps you avoid those pitfalls. Now go forth and conquer those exponents!