Question

1. $sqrt5{a^{2}b^{3}} * sqrt4{a^{2}b}$
2. $sqrt6{x^{2}yz^{3}} * sqrt5{x^{2}yz^{2}}$
3. $sqrt3{x^{2}} * sqrt6{x^{5}}$
4. $sqrt4{a^{3}} * sqrt{a}$
5. $sqrt5{a^{3}b} * sqrt{ab}$
6. $sqrt{a} * sqrt4{a^{3}}$
7. $sqrt5{b^{2}} * sqrt{b^{3}}$
8. $sqrt{xy^{3}} * sqrt3{x^{2}y}$
9. $sqrt6{a^{3}b^{3}} * sqrt4{a^{2}b}$
10. $sqrt3{xy^{2}z} * sqrt4{x^{3}yz^{2}}$

Explanation:

All problems involve converting radicals to rational exponents, multiplying, then simplifying to the simplest radical form.

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Problem 1

Step1: Convert to rational exponents

$a^{\frac{2}{5}}b^{\frac{3}{5}} * a^{\frac{2}{4}}b^{\frac{1}{4}} = a^{\frac{2}{5}}b^{\frac{3}{5}} * a^{\frac{1}{2}}b^{\frac{1}{4}}$

Step2: Add exponents for like bases

$a^{\frac{2}{5}+\frac{1}{2}}b^{\frac{3}{5}+\frac{1}{4}} = a^{\frac{4+5}{10}}b^{\frac{12+5}{20}} = a^{\frac{9}{10}}b^{\frac{17}{20}}$

Step3: Convert back to radicals

$\sqrt[20]{a^{18}b^{17}}$

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Problem 2

Step1: Convert to rational exponents

$x^{\frac{2}{6}}y^{\frac{1}{6}}z^{\frac{3}{6}} * x^{\frac{2}{5}}y^{\frac{1}{5}}z^{\frac{2}{5}} = x^{\frac{1}{3}}y^{\frac{1}{6}}z^{\frac{1}{2}} * x^{\frac{2}{5}}y^{\frac{1}{5}}z^{\frac{2}{5}}$

Step2: Add exponents for like bases

$x^{\frac{1}{3}+\frac{2}{5}}y^{\frac{1}{6}+\frac{1}{5}}z^{\frac{1}{2}+\frac{2}{5}} = x^{\frac{5+6}{15}}y^{\frac{5+6}{30}}z^{\frac{5+4}{10}} = x^{\frac{11}{15}}y^{\frac{11}{30}}z^{\frac{9}{10}}$

Step3: Convert back to radicals

$\sqrt[30]{x^{22}y^{11}z^{27}}$

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Problem 3

Step1: Convert to rational exponents

$x^{\frac{2}{3}} * x^{\frac{5}{6}}$

Step2: Add exponents for x

$x^{\frac{2}{3}+\frac{5}{6}} = x^{\frac{4+5}{6}} = x^{\frac{9}{6}} = x^{\frac{3}{2}}$

Step3: Convert back to radical

$x\sqrt{x}$

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Problem 4

Step1: Convert to rational exponents

$a^{\frac{3}{4}} * a^{\frac{1}{2}}$

Step2: Add exponents for a

$a^{\frac{3}{4}+\frac{1}{2}} = a^{\frac{3+2}{4}} = a^{\frac{5}{4}}$

Step3: Convert back to radical

$a\sqrt[4]{a}$

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Problem 5

Step1: Convert to rational exponents

$a^{\frac{3}{5}}b^{\frac{1}{5}} * a^{\frac{1}{2}}b^{\frac{1}{2}}$

Step2: Add exponents for like bases

$a^{\frac{3}{5}+\frac{1}{2}}b^{\frac{1}{5}+\frac{1}{2}} = a^{\frac{6+5}{10}}b^{\frac{2+5}{10}} = a^{\frac{11}{10}}b^{\frac{7}{10}}$

Step3: Convert back to radical

$a\sqrt[10]{ab^{7}}$

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Problem 6

Step1: Convert to rational exponents

$a^{\frac{1}{2}} * a^{\frac{3}{4}}$

Step2: Add exponents for a

$a^{\frac{1}{2}+\frac{3}{4}} = a^{\frac{2+3}{4}} = a^{\frac{5}{4}}$

Step3: Convert back to radical

$a\sqrt[4]{a}$

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Problem 7

Step1: Convert to rational exponents

$b^{\frac{2}{5}} * b^{\frac{3}{2}}$

Step2: Add exponents for b

$b^{\frac{2}{5}+\frac{3}{2}} = b^{\frac{4+15}{10}} = b^{\frac{19}{10}}$

Step3: Convert back to radical

$b\sqrt[10]{b^{9}}$

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Problem 8

Step1: Convert to rational exponents

$x^{\frac{1}{2}}y^{\frac{3}{2}} * x^{\frac{2}{3}}y^{\frac{1}{3}}$

Step2: Add exponents for like bases

$x^{\frac{1}{2}+\frac{2}{3}}y^{\frac{3}{2}+\frac{1}{3}} = x^{\frac{3+4}{6}}y^{\frac{9+2}{6}} = x^{\frac{7}{6}}y^{\frac{11}{6}}$

Step3: Convert back to radical

$xy\sqrt[6]{xy^{5}}$

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Problem 9

Step1: Convert to rational exponents

$a^{\frac{3}{6}}b^{\frac{3}{6}} * a^{\frac{2}{4}}b^{\frac{1}{4}} = a^{\frac{1}{2}}b^{\frac{1}{2}} * a^{\frac{1}{2}}b^{\frac{1}{4}}$

Step2: Add exponents for like bases

$a^{\frac{1}{2}+\frac{1}{2}}b^{\frac{1}{2}+\frac{1}{4}} = a^{1}b^{\frac{3}{4}}$

Step3: Convert back to radical

$a\sqrt[4]{b^{3}}$

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Problem 10

Step1: Convert to rational exponents

$x^{\frac{1}{3}}y^{\frac{1}{3}}z^{\frac{2}{3}} * x^{\frac{3}{4}}y^{\frac{1}{4}}z^{\frac{2}{4}} = x^{\frac{1}{3}}y^{\frac{1}{3}}z^{\frac{2}{3}} * x^{\frac{3}{4}}y^{\frac{1}{4}}z^{\frac{1}{2}}$

Step2: Add exponents for like bases

$x^{\frac{1}{3}+\frac{3}{4}}y^{\frac{1}{3}+\frac{1}{4}}z^{\frac{2}{3}+\frac{1}{2}} = x^{\frac{4+9}{12}}y^{\frac{4+3}{12}}z^{\frac{4+3}{6}} = x^{\frac{13}{12}}y^{\frac{7}{12}}z^{\frac{7}{6}}$

Step3: Convert back to radical

$xz\sqrt[12]{xy^{7}z^{2}}$

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Answer:

1. $\boldsymbol{\sqrt[20]{a^{18}b^{17}}}$
2. $\boldsymbol{\sqrt[30]{x^{22}y^{11}z^{27}}}$
3. $\boldsymbol{x\sqrt{x}}$
4. $\boldsymbol{a\sqrt[4]{a}}$
5. $\boldsymbol{a\sqrt[10]{ab^{7}}}$
6. $\boldsymbol{a\sqrt[4]{a}}$
7. $\boldsymbol{b\sqrt[10]{b^{9}}}$
8. $\boldsymbol{xy\sqrt[6]{xy^{5}}}$
9. $\boldsymbol{a\sqrt[4]{b^{3}}}$
10. $\boldsymbol{xz\sqrt[12]{xy^{7}z^{2}}}$