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 \(\displaystyle{10}{x}^{{{\frac{{{7}}}{{{12}}}}}}\)

"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 \(\displaystyle{10}{x}^{{{\frac{{{7}}}{{{12}}}}}}\)