nth Root Problems: Problems with Solutions

Solution:
[tex]\sqrt[3]{8} = \sqrt[3]{2^3} = 2[/tex]

Solution:
[tex]\sqrt[3]{-27} = \sqrt[3]{(-3)^3} = -3[/tex]

Solution:
[tex]\sqrt[4]{16} = \sqrt[4]{2^4} = 2[/tex]

Solution:
[tex]\sqrt[4]{81} = \sqrt[4]{3^4} = 3[/tex]

Solution:
[tex]74088=2\times 37044=2^2\times 18522=2^3\times 9261[/tex]
[tex]9261=3\times 3087=3^2\times 1029=3^3\times 343=3^3\times 7^3[/tex]
[tex]74088=2^3\times 3^3\times 7^3=(2\times 3\times 7)^3=42^3[/tex]
Therefore [tex]\sqrt[3]{74088}=42[/tex].

Solution:
[tex]1024=4^5[/tex], [tex]243=3^5[/tex], so [tex]\sqrt[5]{-\frac{1024}{243}}=-\frac{4}{3}[/tex] (an odd root of a negative number is negative).
Convert the mixed number to an improper fraction: [tex]5\frac{1}{16}=\frac{81}{16}=\left(\frac{3}{2}\right)^4[/tex], so [tex]\sqrt[4]{5\frac{1}{16}}=\frac{3}{2}[/tex].
[tex]-\frac{4}{3}\times \frac{3}{2}=-2[/tex]

Solution:
[tex]\sqrt[3]{40}=\sqrt[3]{8\times 5}=2\sqrt[3]{5}[/tex]
[tex]\sqrt[6]{25}=\sqrt[6]{5^2}=\sqrt[3]{5}[/tex] (we divide the index of the root and the exponent by [tex]2[/tex])
[tex]2\sqrt[3]{5}+\sqrt[3]{5}=3\sqrt[3]{5}[/tex]

Solution:
Bring the roots to a common index — the least common multiple of [tex]2[/tex] and [tex]3[/tex], which is [tex]6[/tex]:
[tex]\sqrt{5}=\sqrt[6]{5^3}=\sqrt[6]{125}[/tex], [tex]\sqrt[3]{2}=\sqrt[6]{2^2}=\sqrt[6]{4}[/tex]
[tex]\sqrt{5}\times \sqrt[3]{2}=\sqrt[6]{125\times 4}=\sqrt[6]{500}[/tex]

Solution:
Let [tex]a=\sqrt[3]{11}[/tex] and [tex]b=\sqrt[3]{3}[/tex]. Then [tex]\sqrt[3]{121}=a^2[/tex], [tex]\sqrt[3]{33}=ab[/tex], [tex]\sqrt[3]{9}=b^2[/tex].
We use the sum of cubes formula [tex](a+b)\left(a^2-ab+b^2\right)=a^3+b^3[/tex]:
[tex]a^3+b^3=11+3=14[/tex]

Solution:
A root of even index is non-negative: [tex]\sqrt[4]{a^4}=|a|[/tex].
[tex]\sqrt{5}\lt 3[/tex], so [tex]\sqrt{5}-3\lt 0[/tex] and
[tex]\sqrt[4]{\left(\sqrt{5}-3\right)^4}=\left|\sqrt{5}-3\right|=3-\sqrt{5}[/tex]