Sure, let's find the simplest rationalising factors for each given root step-by-step:
### A) Simplest Rationalising Factor of [tex]\( \sqrt[3]{9} \)[/tex]
To rationalize [tex]\( \sqrt[3]{9} \)[/tex], we need to find a number that, when multiplied by [tex]\( \sqrt[3]{9} \)[/tex], results in a rational number [tex]\(9\)[/tex].
The simplest way to do this is to multiply [tex]\( \sqrt[3]{9} \)[/tex] by [tex]\( \sqrt[3]{9^2} \)[/tex]. When we do this, the exponents add up as follows:
[tex]\[
(\sqrt[3]{9}) \times (\sqrt[3]{9^2}) = 9^{1/3} \times 9^{2/3} = 9^{(1/3 + 2/3)} = 9^1 = 9
\][/tex]
Thus, the rationalising factor of [tex]\( \sqrt[3]{9} \)[/tex] is:
[tex]\[
9^{2/3} \approx 4.3267487109222245
\][/tex]
### B) Simplest Rationalising Factor of [tex]\( \sqrt[4]{3} \)[/tex]
To rationalize [tex]\( \sqrt[4]{3} \)[/tex], we need to multiply [tex]\( \sqrt[4]{3} \)[/tex] by [tex]\( \sqrt[4]{3^3} \)[/tex]. This works similarly:
[tex]\[
(\sqrt[4]{3}) \times (\sqrt[4]{3^3}) = 3^{1/4} \times 3^{3/4} = 3^{(1/4 + 3/4)} = 3^1 = 3
\][/tex]
So, the rationalising factor of [tex]\( \sqrt[4]{3} \)[/tex] is:
[tex]\[
3^{3/4} \approx 2.2795070569547775
\][/tex]
### C) Simplest Rationalising Factor of [tex]\( \sqrt[5]{6} \)[/tex]
To rationalize [tex]\( \sqrt[5]{6} \)[/tex], we need to multiply [tex]\( \sqrt[5]{6} \)[/tex] by [tex]\( \sqrt[5]{6^4} \)[/tex]:
[tex]\[
(\sqrt[5]{6}) \times (\sqrt[5]{6^4}) = 6^{1/5} \times 6^{4/5} = 6^{(1/5 + 4/5)} = 6^1 = 6
\][/tex]
So, the rationalising factor of [tex]\( \sqrt[5]{6} \)[/tex] is:
[tex]\[
6^{4/5} \approx 4.192962712629476
\][/tex]
### D) Simplest Rationalising Factor of [tex]\( \sqrt[6]{7} \)[/tex]
For [tex]\( \sqrt[6]{7} \)[/tex], we should multiply [tex]\( \sqrt[6]{7} \)[/tex] by [tex]\( \sqrt[6]{7^5} \)[/tex]:
[tex]\[
(\sqrt[6]{7}) \times (\sqrt[6]{7^5}) = 7^{1/6} \times 7^{5/6} = 7^{(1/6 + 5/6)} = 7^1 = 7
\][/tex]
Hence, the rationalising factor of [tex]\( \sqrt[6]{7} \)[/tex] is:
[tex]\[
7^{5/6} \approx 5.061140184796387
\][/tex]
Thus, the simplest rationalising factors are:
- [tex]\( \sqrt[3]{9} \)[/tex]: [tex]\(4.3267487109222245\)[/tex]
- [tex]\( \sqrt[4]{3} \)[/tex]: [tex]\(2.2795070569547775\)[/tex]
- [tex]\( \sqrt[5]{6} \)[/tex]: [tex]\(4.192962712629476\)[/tex]
- [tex]\( \sqrt[6]{7} \)[/tex]: [tex]\(5.061140184796387\)[/tex]
