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})

Rational exponents and radicals
ANSWERED
asked 2020-12-01
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) \(\displaystyle{\left({5}\sqrt{{{3}}}{\left\lbrace{x}\right\rbrace}\right)}{\left({2}\sqrt{{{4}}}{\left\lbrace{x}\right\rbrace}\right)}\)

Answers (1)

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