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Rational exponents can sometimes seem tricky, but they're simply another way of writing roots and powers. For example, the exponent \(\frac{3}{2}\) means both cubing and taking the square root.
So, if we have \((x + 2)^{\frac{3}{2}}\), it's the same as taking the square root of \(x + 2\) and then cubing the result:
\((x + 2)^{\frac{3}{2}} = \left((x + 2)^{\frac{1}{2}}\right)^3\). \ Exponents like these are common in algebra and understanding them will greatly help in solving equations.
Rational exponents follow the same rules as integer exponents; for example:

• \(a^m \cdot a^n = a^{m+n}\)
• \(\left(a^m\right)^n = a^{m \cdot n}\)

A key insight in solving equations with exponents is understanding the base. The base of an exponent is the number that's being raised to a power.
In the equation \((x + 2)^{\frac{3}{2}} \), the base is \(x + 2\). Now, it's important to know that for rational exponents like \(\frac{3}{2}\), the base must be non-negative when dealing with real numbers.
This is because you can't take the even root (like square root) of a negative number without involving imaginary numbers.
So, if \((x + 2)^{\frac{3}{2}} = -1\), there's no real solution because a non-negative number raised to a power will never result in a negative number.
This is why Step 2 in the given solution demonstrates the impossibility of finding a real number solution.

When solving equations, knowing if you're targeting real numbers is critical. Real numbers include all the numbers on the number line, including positives, negatives, and zero.
However, they do not include imaginary or complex numbers. In the exercise given, we're asked to solve \((x + 2)^{\frac{3}{2}} = -1\) for real numbers.
Since the base \(x + 2\) should be non-negative and cannot produce a negative result when raised to a power, there's no real \(x\) satisfying this equation.
That’s why the conclusion in Step 3 states there's no real solution. It's essential to distinguish the domain of your solution - whether you're finding solutions in real numbers, complex numbers, or another set.
This clarity helps avoid pitfalls and ensures you apply the right rules and properties to your solutions.