To find the cube root of 4, which can be expressed as [tex]\( \sqrt[3]{4} \)[/tex] or [tex]\( 4^{\frac{1}{3}} \)[/tex], we can follow a detailed step-by-step approach.
1. Understand the Expression:
The expression [tex]\( 4^{\frac{1}{3}} \)[/tex] indicates we are looking for a number that, when raised to the power of 3 (cubed), equals 4.
2. Set Up the Equation:
We want to solve for [tex]\( x \)[/tex] in the equation:
[tex]\[ x^3 = 4 \][/tex]
3. Rewrite in Exponential Form:
To solve for [tex]\( x \)[/tex], we can rewrite it using the exponent notation [tex]\( x = 4^{\frac{1}{3}} \)[/tex].
4. Determining the Value:
To determine the value of [tex]\( 4^{\frac{1}{3}} \)[/tex], we can use the property of exponents and the known results.
5. Approximate Evaluation:
Through mathematical analysis and approximation techniques, we find that:
[tex]\[ 4^{\frac{1}{3}} \approx 1.5874010519681994 \][/tex]
Thus, the cube root of 4 is approximately:
[tex]\[ \sqrt[3]{4} = 1.5874010519681994 \][/tex]
This result tells us that 1.5874010519681994 is the number, which, when cubed, gives a result very close to 4.
