Sure, let's solve this step-by-step by simplifying each term one by one:
1. Expression 1:
[tex]\[
3 b^2 \left(\sqrt[3]{54 a}\right)
\][/tex]
Using the property of cube roots, we have:
[tex]\[
\sqrt[3]{54 a} = (54 a)^{1/3}
\][/tex]
2. Expression 2:
[tex]\[
3 \left(\sqrt[3]{2 a b^6}\right)
\][/tex]
This expression can be broken down as:
[tex]\[
\sqrt[3]{2 a b^6} = (2 a b^6)^{1/3} = 2^{1/3} a^{1/3} b^2
\][/tex]
3. Expression 3:
[tex]\[
8 b^2 \left(\sqrt[3]{2 a}\right)
\][/tex]
Simplifying this:
[tex]\[
\sqrt[3]{2 a} = (2 a)^{1/3}
\][/tex]
4. Expression 4:
[tex]\[
12 h^2 \left(\sqrt[3]{2 d}\right)
\][/tex]
Simplifying this:
[tex]\[
\sqrt[3]{2 d} = (2 d)^{1/3}
\][/tex]
5. Expression 5:
[tex]\[
6 b^2 \left(\sqrt[8]{2 a}\right)
\][/tex]
Simplifying this:
[tex]\[
\sqrt[8]{2 a} = (2 a)^{1/8}
\][/tex]
6. Expression 6:
[tex]\[
12 h^2 \left(8 \sqrt{2 d}\right)
\][/tex]
Calculate the inner part first:
[tex]\[
8 \sqrt{2 d} = 8 (2 d)^{1/2}
\][/tex]
Simplifying:
[tex]\[
8 (2 d)^{1/2} = 8 \cdot 2^{1/2} d^{1/2} = 8 \sqrt{2} \sqrt{d}
\][/tex]
Now we sum up all the simplified expressions:
[tex]\[
3 b^2 \left(\sqrt[3]{54 a}\right) + 3 \left(2^{1/3} a^{1/3} b^2\right) + 8 b^2 \left(2^{1/3} a^{1/3}\right) + 12 h^2 \left(2^{1/3} d^{1/3}\right) + 6 b^2 \left(2^{1/8} a^{1/8}\right) + 12 h^2 \left(8 \sqrt{2} \sqrt{d}\right)
\][/tex]
Now gather all the simplified results:
1. [tex]\[ 3b^2(54a)^{1/3} \][/tex]
2. [tex]\[ 3 \cdot 2^{1/3} a^{1/3} b^2 \][/tex]
3. [tex]\[ 8 b^2 \cdot 2^{1/3} a^{1/3} \][/tex]
4. [tex]\[ 12h^2 \cdot 2^{1/3} d^{1/3} \][/tex]
5. [tex]\[ 6 b^2 \cdot 2^{1/8} a^{1/8} \][/tex]
6. [tex]\[ 12h^2 \cdot 8 \sqrt{2} \sqrt{d} \][/tex]
By combining all these together, we get the total result:
[tex]\[
3 b^2 (54 a)^{1/3} + 3 \cdot 2^{1/3} a^{1/3} b^2 + 8 \cdot 2^{1/3} b^2 a^{1/3} + 12 \cdot 2^{1/3} h^2 d^{1/3} + 6 \cdot 2^{1/8} b^2 a^{1/8} + 12 \cdot 8 \cdot \sqrt{2} h^2 \sqrt{d}
\][/tex]
[tex]\[
\boxed{3 b^2 (54 a)^{1/3} + 3 \cdot 2^{1/3} a^{1/3} b^2 + 8 \cdot 2^{1/3} b^2 a^{1/3} + 12 \cdot 2^{1/3} h^2 d^{1/3} + 6 \cdot 2^{1/8} b^2 a^{1/8} + 12 \cdot 8 \cdot \sqrt{2} h^2 \sqrt{d}}
\][/tex]
