To solve the expression [tex]\(5^{-\frac{3}{4}} \times 5^{\frac{1}{4}}\)[/tex], let's go through the steps in detail.
1. Separate the terms:
- The first term is [tex]\(5^{-\frac{3}{4}}\)[/tex].
- The second term is [tex]\(5^{\frac{1}{4}}\)[/tex].
2. Evaluate each term separately:
- [tex]\(5^{-\frac{3}{4}} \approx 0.2990697562442441\)[/tex].
- [tex]\(5^{\frac{1}{4}} \approx 1.4953487812212205\)[/tex].
3. Multiply the two terms together:
- Multiply [tex]\(0.2990697562442441\)[/tex] by [tex]\(1.4953487812212205\)[/tex].
- The product is approximately [tex]\(0.44721359549995787\)[/tex].
4. Simplify using the properties of exponents:
- According to the property of exponents, [tex]\(a^m \times a^n = a^{m+n}\)[/tex].
- Combine the exponents: [tex]\(-\frac{3}{4} + \frac{1}{4} = -\frac{2}{4} = -\frac{1}{2}\)[/tex].
5. Recalculate and verify the simplified result:
- Simplify the base and the new exponent: [tex]\(5^{-\frac{1}{2}} = \frac{1}{\sqrt{5}} = 5^{-0.5} \approx 0.44721359549995787\)[/tex].
Therefore, the final answer for the expression [tex]\(5^{-\frac{3}{4}} \times 5^{\frac{1}{4}}\)[/tex] is approximately [tex]\(0.44721359549995787\)[/tex].
