To rewrite the expression [tex]\((2 \cdot 6)^{\frac{3}{2}}\)[/tex] using the properties of rational exponents:
1. Multiply the bases: First, calculate the product of the bases:
[tex]\[
2 \cdot 6 = 12
\][/tex]
2. Rewrite the expression: Replace [tex]\(2 \cdot 6\)[/tex] with the result from step 1:
[tex]\[
(2 \cdot 6)^{\frac{3}{2}} = 12^{\frac{3}{2}}
\][/tex]
3. Simplify using rational exponent properties: To simplify [tex]\(12^{\frac{3}{2}}\)[/tex], express it as a combination of root and power. The rational exponent [tex]\(\frac{3}{2}\)[/tex] means:
[tex]\[
12^{\frac{3}{2}} = \left(12^{\frac{1}{2}}\right)^3
\][/tex]
[tex]\[
12^{\frac{1}{2}} = \sqrt{12}
\][/tex]
4. Calculate the square root: The square root of 12 can be further simplified, but since we are looking for the final simplified form in terms of decimal approximation:
[tex]\[
\sqrt{12} \approx 3.4641
\][/tex]
5. Raise to the power of 3: Finally, raise the result to the third power:
[tex]\[
(3.4641)^3 \approx 41.569
\][/tex]
Therefore, the expression [tex]\((2 \cdot 6)^{\frac{3}{2}}\)[/tex] in its simplest form is approximately:
[tex]\[
41.569
\][/tex]
