Answer :
To simplify the expression [tex]\(\sqrt[3]{2} \times \sqrt{5}\)[/tex], let's begin by breaking down each term and examining how they interact.
### Step-by-Step Solution:
1. Understand the individual components:
- We have two distinct radicals to work with:
[tex]\[ \sqrt[3]{2} \quad \text{and} \quad \sqrt{5} \][/tex]
- [tex]\(\sqrt[3]{2}\)[/tex] represents the cube root of 2.
- [tex]\(\sqrt{5}\)[/tex] represents the square root of 5.
2. Express the radicals with fractional exponents:
- Recall that [tex]\(\sqrt[3]{2}\)[/tex] can be written as [tex]\(2^{1/3}\)[/tex].
- Similarly, [tex]\(\sqrt{5}\)[/tex] can be written as [tex]\(5^{1/2}\)[/tex].
Therefore, our expression can be rewritten as:
[tex]\[ 2^{1/3} \times 5^{1/2} \][/tex]
3. Combine the exponents and multiply:
- These terms have different bases (2 and 5) and thus cannot be combined into a single exponent. However, we can multiply these decimal approximations directly:
[tex]\[ 2^{1/3} \approx 1.2599 \quad \text{and} \quad 5^{1/2} \approx 2.2361 \][/tex]
4. Multiply the numerical approximations:
- Multiplying their decimal approximations gives us an approximate numerical value:
[tex]\[ 1.2599 \times 2.2361 \approx 2.8173 \][/tex]
So, [tex]\(\sqrt[3]{2} \times \sqrt{5} \approx 2.8173\)[/tex].
Therefore, the simplified form of the expression [tex]\(\sqrt[3]{2} \times \sqrt{5}\)[/tex] is approximately [tex]\(2.8173\)[/tex].
### Step-by-Step Solution:
1. Understand the individual components:
- We have two distinct radicals to work with:
[tex]\[ \sqrt[3]{2} \quad \text{and} \quad \sqrt{5} \][/tex]
- [tex]\(\sqrt[3]{2}\)[/tex] represents the cube root of 2.
- [tex]\(\sqrt{5}\)[/tex] represents the square root of 5.
2. Express the radicals with fractional exponents:
- Recall that [tex]\(\sqrt[3]{2}\)[/tex] can be written as [tex]\(2^{1/3}\)[/tex].
- Similarly, [tex]\(\sqrt{5}\)[/tex] can be written as [tex]\(5^{1/2}\)[/tex].
Therefore, our expression can be rewritten as:
[tex]\[ 2^{1/3} \times 5^{1/2} \][/tex]
3. Combine the exponents and multiply:
- These terms have different bases (2 and 5) and thus cannot be combined into a single exponent. However, we can multiply these decimal approximations directly:
[tex]\[ 2^{1/3} \approx 1.2599 \quad \text{and} \quad 5^{1/2} \approx 2.2361 \][/tex]
4. Multiply the numerical approximations:
- Multiplying their decimal approximations gives us an approximate numerical value:
[tex]\[ 1.2599 \times 2.2361 \approx 2.8173 \][/tex]
So, [tex]\(\sqrt[3]{2} \times \sqrt{5} \approx 2.8173\)[/tex].
Therefore, the simplified form of the expression [tex]\(\sqrt[3]{2} \times \sqrt{5}\)[/tex] is approximately [tex]\(2.8173\)[/tex].
