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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].
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