Introduction

In the realm of mathematics, exploring the relationships between square roots and cube roots can lead to fascinating discoveries. This article will delve into a specific problem involving the square root being half of the cube root. We will begin by establishing the equation, solving it step by step, and providing examples to enhance understanding.

Establishing the Equation

To find a number x where its square root is half of its cube root, we start with the equation:

sqrt{x} frac{1}{2} sqrt[3]{x}

Step 1: Eliminating the Square Root and Cube Root

Our first step is to eliminate the square root and cube root by squaring both sides:

sqrt{x}^2 left(frac{1}{2} sqrt[3]{x}right)^2

This simplifies to:

x frac{1}{4} sqrt[3]{x}^2

Step 2: Expressing the Cube Root as a Fractional Exponent

Next, we express sqrt[3]{x} as x^{1/3}:

x frac{1}{4} (x^{1/3})^2

This simplifies to:

x frac{1}{4} x^{2/3}

Step 3: Multiplying Both Sides by 4

To eliminate the fraction, we multiply both sides by 4:

4x x^{2/3}

Next, we rearrange the equation:

x^{2/3} - 4x 0

Factoring out x:

x(x^{-2/3} - 4) 0

This gives us two solutions:

x 0 x^{-1/3} - 4 0

Step 4: Solving for the Second Solution

For the second solution, we solve:

x^{-1/3} 4

Taking the reciprocal:

x^{1/3} frac{1}{4}

Cubing both sides:

x left(frac{1}{4}right)^3 frac{1}{64}

Thus, the numbers that satisfy the original condition are:

boxed{0} quad and quad boxed{frac{1}{64}}

Further Exploration

Example 1: Taking Both Sides to the Sixth Power

As another approach, we can take both sides to the sixth power:

(sqrt{x})^6 left(frac{1}{2} (x^{1/3})right)^6

This simplifies to:

x^3 frac{1}{64} x^2

Multiplying both sides by 64:

64x^3 x^2

Subtracting both sides by (x^2):

64x^3 - x^2 0

Factoring out x^2:

x^2(64x - 1) 0

Giving us the solutions:

x^2 0 Rightarrow x 0 64x - 1 0 Rightarrow 64x 1 Rightarrow x frac{1}{64}

Example 2: Finding a Number Where the Square Root is Double of the Cube Root

Now, let's find a number where the square root is double of its cube root:

sqrt{x} 2 sqrt[3]{x}

Cubing both sides:

x (2x^{1/3})^3

This simplifies to:

x 8x

Subtracting both sides by (x):

x - 8x 0 Rightarrow -7x 0 Rightarrow x 64

Thus, the number is:

64 (√64 8 and 3√64 4)

Conclusion

In conclusion, through these mathematical explorations, we have demonstrated how to solve equations involving square roots and cube roots. The key to solving such problems is careful manipulation and understanding of the properties of exponents.