Alright! Let's simplify the expression [tex]\(10^{-1 / 3} \times 25^{2 / 3} \times 2^{5 / 3}\)[/tex] step by step using prime factors.
1. Factorize the components into their prime factors:
- [tex]\(10 = 2 \times 5\)[/tex]
- [tex]\(25 = 5^2\)[/tex]
- [tex]\(2\)[/tex] is already a prime number.
So the expression can be rewritten as:
[tex]\[
(2 \times 5)^{-1 / 3} \times (5^2)^{2 / 3} \times 2^{5 / 3}
\][/tex]
2. Apply the exponents to the factors individually:
- For [tex]\( (2 \times 5)^{-1 / 3} \)[/tex]:
[tex]\[
(2 \times 5)^{-1 / 3} = 2^{-1 / 3} \times 5^{-1 / 3}
\][/tex]
- For [tex]\( (5^2)^{2 / 3} \)[/tex]:
[tex]\[
(5^2)^{2 / 3} = 5^{4 / 3}
\][/tex]
- For [tex]\( 2^{5 / 3} \)[/tex], it remains as:
[tex]\[
2^{5 / 3}
\][/tex]
Now, combine all simplified components together:
[tex]\[
2^{-1 / 3} \times 5^{-1 / 3} \times 5^{4 / 3} \times 2^{5 / 3}
\][/tex]
3. Combine like bases by adding their exponents:
- For the base [tex]\(2\)[/tex]:
[tex]\[
2^{-1 / 3} \times 2^{5 / 3} = 2^{(-1 / 3 + 5 / 3)}
\][/tex]
Simplify the exponent:
[tex]\[
2^{(-1 / 3 + 5 / 3)} = 2^{4 / 3}
\][/tex]
- For the base [tex]\(5\)[/tex]:
[tex]\[
5^{-1 / 3} \times 5^{4 / 3} = 5^{(-1 / 3 + 4 / 3)}
\][/tex]
Simplify the exponent:
[tex]\[
5^{(-1 / 3 + 4 / 3)} = 5^{3 / 3} = 5^1 = 5
\][/tex]
4. Combine the results:
Now we have:
[tex]\[
2^{4 / 3} \times 5
\][/tex]
5. Calculate the numerical value:
Calculate [tex]\(2^{4 / 3}\)[/tex]:
[tex]\[
2^{4 / 3} \approx 2.5198421
\][/tex]
So,
[tex]\[
2^{4 / 3} \times 5 \approx 2.5198421 \times 5 = 12.5992105
\][/tex]
Therefore, the simplified expression evaluates to approximately [tex]\(12.5992105\)[/tex].
