To solve [tex]\(\sqrt[4]{3} \times \sqrt[4]{27}\)[/tex], we need to simplify and combine the expressions step-by-step. Follow along carefully:
1. Expressing both terms using fractional exponents:
[tex]\[
\sqrt[4]{3} = 3^{1/4}
\][/tex]
[tex]\[
\sqrt[4]{27} = 27^{1/4}
\][/tex]
2. Rewrite 27 using its prime factorization:
[tex]\[
27 = 3 \times 3 \times 3 = 3^3
\][/tex]
Substituting in this factorization, we get:
[tex]\[
\sqrt[4]{27} = (3^3)^{1/4}
\][/tex]
3. Applying the power of a power property [tex]\((a^{m})^n = a^{m \times n}\)[/tex]:
For the expression [tex]\((3^3)^{1/4}\)[/tex]:
[tex]\[
(3^3)^{1/4} = 3^{3 \times (1/4)} = 3^{3/4}
\][/tex]
4. Now substitute back into our original expression:
[tex]\[
\sqrt[4]{3} \times \sqrt[4]{27} = 3^{1/4} \times 3^{3/4}
\][/tex]
5. Using the property of exponents [tex]\((a^m \times a^n = a^{m+n})\)[/tex] to combine the terms:
[tex]\[
3^{1/4} \times 3^{3/4} = 3^{1/4 + 3/4}
\][/tex]
Simplify the exponent:
[tex]\[
1/4 + 3/4 = 4/4 = 1
\][/tex]
So we have:
[tex]\[
3^1 = 3
\][/tex]
Hence, the simplified result of [tex]\(\sqrt[4]{3} \times \sqrt[4]{27}\)[/tex] is [tex]\(3\)[/tex].
To provide the intermediate results from each step in numerical form:
- [tex]\(\sqrt[4]{3} \approx 1.3160740129524924\)[/tex]
- [tex]\(\sqrt[4]{27} = (3^3)^{1/4} \approx 2.2795070569547775\)[/tex]
- The product [tex]\(1.3160740129524924 \times 2.2795070569547775 \approx 2.9999999999999996\)[/tex]
Thus, the final result is [tex]\(3\)[/tex].
