Let's solve the problem step-by-step.
We are given the expression [tex]\( 24^{\frac{1}{3}} \)[/tex]. This expression implies we need to find the cube root of 24. The cube root of a number [tex]\( x \)[/tex] is a value that, when raised to the power of 3, yields [tex]\( x \)[/tex]. Mathematically, this is written as [tex]\( x^{\frac{1}{3}} \)[/tex].
Now, let's examine the given options to see which one matches [tex]\( 24^{\frac{1}{3}} \)[/tex]:
1. [tex]\( 2 \sqrt{3} \)[/tex]: This expression means 2 times the square root of 3. The square root is different from the cube root, so this does not match [tex]\( 24^{\frac{1}{3}} \)[/tex].
2. [tex]\( 2 \sqrt[3]{3} \)[/tex]: This means 2 times the cube root of 3. To verify this, we take [tex]\( 2 \cdot 3^{\frac{1}{3}} \)[/tex].
3. [tex]\( 2 \sqrt{6} \)[/tex]: This means 2 times the square root of 6. Again, the square root is different from the cube root.
4. [tex]\( 2 \sqrt[3]{6} \)[/tex]: This means 2 times the cube root of 6. To verify this, we take [tex]\( 2 \cdot 6^{\frac{1}{3}} \)[/tex].
Among these options, we are checking to see which one matches the provided value of the cube root of 24.
The calculated value of [tex]\( 24^{\frac{1}{3}} \)[/tex] is approximately [tex]\( 2.8845 \)[/tex]. Let's compare this with each option by approximating:
1. [tex]\( 2 \sqrt{3} \approx 2 \cdot 1.732 = 3.464 \)[/tex]
2. [tex]\( 2 \sqrt[3]{3} \approx 2 \cdot 1.442 = 2.884 \)[/tex] (This is very close to 2.8845)
3. [tex]\( 2 \sqrt{6} \approx 2 \cdot 2.449 = 4.898 \)[/tex]
4. [tex]\( 2 \sqrt[3]{6} \approx 2 \cdot 1.817 = 3.634 \)[/tex]
From these comparisons, you can see that the option [tex]\( 2 \sqrt[3]{3} \)[/tex] ([tex]\( 2 \cdot 3^{1/3} \)[/tex]) is the closest match to [tex]\( 24^{1/3} \)[/tex].
Thus, the expression that is equivalent to [tex]\( 24^{\frac{1}{3}} \)[/tex] is:
[tex]\[ \boxed{2 \sqrt[3]{3}} \][/tex]
