Sure! Let's simplify each expression step by step.
1. Simplifying [tex]\(\sqrt{8}\)[/tex]:
[tex]\[
\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2 \times \sqrt{2}
\][/tex]
The simplified form is [tex]\(2\sqrt{2}\)[/tex], and numerically it is approximately:
[tex]\[
2 \cdot \sqrt{2} \approx 2.8284271247461903
\][/tex]
2. Simplifying [tex]\(\sqrt{48}\)[/tex]:
[tex]\[
\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4 \times \sqrt{3}
\][/tex]
The simplified form is [tex]\(4\sqrt{3}\)[/tex], and numerically it is approximately:
[tex]\[
4 \cdot \sqrt{3} \approx 6.928203230275509
\][/tex]
3. Simplifying [tex]\(3\sqrt{294}\)[/tex]:
[tex]\[
3\sqrt{294} = 3\sqrt{49 \times 6} = 3\sqrt{49} \times \sqrt{6} = 3 \times 7 \times \sqrt{6} = 21 \times \sqrt{6}
\][/tex]
The simplified form is [tex]\(21\sqrt{6}\)[/tex], and numerically it is approximately:
[tex]\[
21 \cdot \sqrt{6} \approx 51.43928459844673
\][/tex]
4. Simplifying [tex]\(\sqrt[3]{16}\)[/tex]:
[tex]\[
\sqrt[3]{16} = \sqrt[3]{8 \times 2} = \sqrt[3]{8} \times \sqrt[3]{2} = 2 \times \sqrt[3]{2}
\][/tex]
The simplified form is [tex]\(2\sqrt[3]{2}\)[/tex], and numerically it is approximately:
[tex]\[
2 \cdot \sqrt[3]{2} \approx 2.5198420997897464
\][/tex]
5. Simplifying [tex]\(\sqrt[3]{135}\)[/tex]:
[tex]\[
\sqrt[3]{135} = \sqrt[3]{27 \times 5} = \sqrt[3]{27} \times \sqrt[3]{5} = 3 \times \sqrt[3]{5}
\][/tex]
The simplified form is [tex]\(3\sqrt[3]{5}\)[/tex], and numerically it is approximately:
[tex]\[
3 \cdot \sqrt[3]{5} \approx 5.12992784003009
\][/tex]
Thus, the simplified forms and their numerical approximations are:
1. [tex]\(2\sqrt{2} \approx 2.8284271247461903\)[/tex]
2. [tex]\(4\sqrt{3} \approx 6.928203230275509\)[/tex]
3. [tex]\(21\sqrt{6} \approx 51.43928459844673\)[/tex]
4. [tex]\(2\sqrt[3]{2} \approx 2.5198420997897464\)[/tex]
5. [tex]\(3\sqrt[3]{5} \approx 5.12992784003009\)[/tex]
