Question

# Radical simplify the expression, and eliminate any negative exponents(s). Assume that all letters denote positive numbers. a) sqrt[6]{y^{5}} sqrt[3]{y^{2}} b) (5sqrt[3]{x})(2sqrt[4]{x})

Radical simplify the expression, and eliminate any negative exponents(s). Assume that all letters denote positive numbers.
a) $$\displaystyle\sqrt{{{6}}}{\left\lbrace{y}^{{{5}}}\right\rbrace}\sqrt{{{3}}}{\left\lbrace{y}^{{{2}}}\right\rbrace}$$
b) $$\displaystyle{\left({5}\sqrt{{{3}}}{\left\lbrace{x}\right\rbrace}\right)}{\left({2}\sqrt{{{4}}}{\left\lbrace{x}\right\rbrace}\right)}$$

2020-12-02
a) Formula used:
"The product of two powers with the same base a and different exponents m and n is given by, $$\displaystyle{a}^{{{m}}}\times{a}^{{{n}}}={a}^{{{m}+{n}}}.$$
That is, while multiplying two powers with same base, the exponents are added and the base will remain the same.
Calculation:
The given expression is $$\displaystyle\sqrt{{{6}}}{\left\lbrace{y}^{{{5}}}\right\rbrace}\sqrt{{{3}}}{\left\lbrace{y}^{{{2}}}\right\rbrace}.$$
Use the definition of rational exponents and the above formula and sinplify the expression as shown below.
$$\displaystyle\sqrt{{{6}}}{\left\lbrace{y}^{{{5}}}\right\rbrace}\sqrt{{{3}}}{\left\lbrace{y}^{{{2}}}\right\rbrace}={y}^{{{\frac{{{5}}}{{{6}}}}}}\times{y}^{{{\frac{{{2}}}{{{3}}}}}}$$
$$\displaystyle={y}^{{{\frac{{{5}}}{{{6}}}}+{\frac{{{2}}}{{{3}}}}}}$$
$$\displaystyle={y}^{{{\frac{{{3}}}{{{2}}}}}}$$
Therefore, the simplified form of the expression is $$\displaystyle={y}^{{{\frac{{{3}}}{{{2}}}}}}.$$
b) Use the definition of rational exponents and the formula mentioned in sub part (a) and simplify the expression as shown below.
$$\displaystyle{\left({5}\sqrt{{{3}}}{\left\lbrace{x}\right\rbrace}\right)}{\left({2}\sqrt{{{4}}}{\left\lbrace{x}\right\rbrace}\right)}={\left({5}{x}^{{{\frac{{{1}}}{{{3}}}}}}\right)}{\left({2}{x}^{{{\frac{{{1}}}{{{4}}}}}}\right)}$$
$$\displaystyle={10}{x}^{{{\frac{{{1}}}{{{3}}}}+{\frac{{{1}}}{{{4}}}}}}$$
$$\displaystyle={10}{x}^{{{\frac{{{7}}}{{{12}}}}}}$$
Therefore, the simplified form of the expression is $$\displaystyle{10}{x}^{{{\frac{{{7}}}{{{12}}}}}}$$