Solving with Exponentials: Unveiling the Number When Its Cube Root's Square Root is 2

Mathematics often reveals its mysteries through a series of logical steps. One such intriguing problem is determining the number when its cube root's square root is 2. This article explores how to solve such an equation using exponentials.

Understanding the Problem

The problem gives us a relationship between a number, let's call it (x), and the operation of taking the square root and the cube root. Specifically, it states:

Given: ( sqrt[3]{sqrt{x}} 2 )
Find: x

To tackle this, we'll break it down into manageable steps using exponentials and properties of roots and powers.

Solving the Exponential Equation

Starting with the given equation:

( sqrt[3]{sqrt{x}} 2 ) Recall that the square root of a number can be expressed as a fractional exponent. Therefore, we can write: ( sqrt{x} x^{1/2} ) Substitute this back into the original equation: ( sqrt[3]{x^{1/2}} 2 ) Next, we need to express the cube root as a fractional exponent. So: ( x^{1/2} ) becomes: ( x^{1/2 times 1/3} x^{1/6} ) Thus, the equation now looks like: ( x^{1/6} 2 ) To eliminate the exponent, we take the 6th power of both sides: ( (x^{1/6})^6 2^6 ) This simplifies to: ( x 2^6 ) Calculate the power on the right side: ( x 64 )

Therefore, the number is 64.

Double-Checking the Solution

To verify the solution, let's follow the chain of operations backwards.

Start with the solution: ( x 64 ) First, take the cube root of 64: ( sqrt[3]{64} 4 ) Then, take the square root of the result: ( sqrt{4} 2 ) This confirms that our solution is correct.

In conclusion, by breaking down the problem into its basic components and leveraging exponentials, we successfully determined that the number is 64 when its cube root's square root is 2.

Conclusion

Understanding and solving exponential equations can be quite powerful. Whether you're working with roots or higher powers, the key is to express everything in terms of exponents and then simplify stepwise. This method not only helps in solving such problems but also enhances one's problem-solving skills in mathematics.

Feel free to try similar problems or share your thoughts in the comments section below!