$\left(3 k^2\right)^{\frac{1}{3}}=\left(\sqrt[3]{3 k^2}\right)$

Question

04-11-2026
Mathematics

Answered

$\left(3 k^2\right)^{\frac{1}{3}}=\left(\sqrt[3]{3 k^2}\right)$

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-04T15:54:27-05:00

The expression $(3k^2)^{\frac{1}{3}} = \sqrt[3]{3k^2}$ follows the fundamental property of exponents, where $x^{\frac{1}{n}} = \sqrt[n]{x}$.
By substituting $x = 3k^2$ and $n = 3$ into this definition, we see the left-hand side is identically equal to the right-hand side.
This identity is valid for all real numbers $k$, confirming the mathematical consistency of the given equation.
The statement is $\boxed{\text{True}}$.

Explanation

Analyzing the Expression

The problem presents the equation $(3k^2)^{\frac{1}{3}} = \sqrt[3]{3k^2}$ and asks us to confirm its mathematical validity. We analyze this expression by identifying it as a specific case of the general rules governing exponents and radicals. For any real number $x$ and any positive integer $n$, a fractional exponent can be expressed as a radical according to the definition $x^{\frac{1}{n}} = \sqrt[n]{x}$.

Verifying the Identity

To evaluate the identity $(3k^2)^{\frac{1}{3}} = \sqrt[3]{3k^2}$, we set the base $x = 3k^2$ and the index $n = 3$. According to the radical definition, the expression on the left-hand side $(3k^2)^{\frac{1}{3}}$ is equivalent to taking the cube root of the base $3k^2$. This is precisely what is written on the right-hand side as $\sqrt[3]{3k^2}$. Thus, the equation is a direct application of the definition of fractional exponents and is mathematically consistent for all real numbers $k$.

Conclusion

Since the transformation between the exponential form and the radical form follows the standard definition, the equality $(3k^2)^{\frac{1}{3}} = \sqrt[3]{3k^2}$ is an identity that holds true for every real value of $k$. We have confirmed the mathematical consistency of the given statement. The final result is simply the confirmation of the identity: $\boxed{\text{True}}$.

Examples

Understanding the relationship between fractional exponents and radicals is essential in fields like physics and engineering. For example, if you are calculating the volume of a sphere given its radius, the formula involves cubing the radius, but if you need to determine the side length of a cube given its volume $V$, you must calculate $V^{1/3}$ or $\sqrt[3]{V}$. Recognizing that these two notations are equivalent allows scientists and engineers to simplify complex formulas and perform calculations more efficiently, regardless of whether they prefer using exponents or root symbols.