(a) Definition used:

Definition of rational exponents:

"For any rational exponent m/n in its lowest terms, where m and n are integers and \(\displaystyle{n}>{0}\ \text{we define,}\ \displaystyle{\left({a}\right)}^{{\frac{m}{{n}}}}={\sqrt[{{n}}]{{{\left({a}\right)}^{m}}}}={\left({\sqrt[{{n}}]{{{a}}}}\right)}^{m}.\)

Formula used:

Laws of exponents:

"To raise a power to a new power, multiply the exponents".

\(\displaystyle{\left({a}^{m}\right)}^{n}={a}^{{{m}{n}}}\)

Calculation:

The given expression is \(\displaystyle{27}^{{\frac{1}{{3}}}}.\)

Use the above mentioned definition and simplify the expression as shown below.

\(\displaystyle{27}^{{\frac{1}{{3}}}}={\left({3}\times{3}\times{3}\right)}^{{\frac{1}{{3}}}}\)

\(\displaystyle={\left({3}^{3}\right)}^{{\frac{1}{{3}}}}\)

\(\displaystyle={3}^{{{3}\times\frac{1}{{3}}}}\)

\(= 3\)

Thus, the value of the expression \(\displaystyle{27}^{{\frac{1}{{3}}}}\) is 3.

(b) Use the above definition and formula mentioned in sub part (a) and simplify the expression as shown below.

\(\displaystyle{\left(-{8}\right)}^{{\frac{1}{{3}}}}={\left({\left(-{2}\right)}\times{\left(-{2}\right)}\times{\left(-{2}\right)}\right)}^{{\frac{1}{{3}}}}\)

\(\displaystyle={\left(-{2}\right)}^{{{3}\times\frac{1}{{3}}}}\)

\(\displaystyle={\left(-{2}\right)}^{{{3}\times\frac{1}{{3}}}}\)

\(=\ - 2\)

Thus, the value of the expression \(\displaystyle{\left(-{8}\right)}^{{\frac{1}{{3}}}}\) is (-2).

(c) Use the above definition and formula mentioned in sub part (a) and simplify the expression as shown below.

\(\displaystyle-{\left(\frac{1}{{8}}\right)}^{{\frac{1}{{3}}}}=-{\left(\frac{1}{{2}^{3}}\right)}^{{\frac{1}{{3}}}}\)

\(\displaystyle=-{\left({2}^{ -{{3}}}\right)}^{{\frac{1}{{3}}}}\)

\(\displaystyle=-{\left({2}^{{-{3}\times\frac{1}{{3}}}}\right)}\)

On further simplifications, the following is obtained.

\(\displaystyle-{\left(\frac{1}{{8}}\right)}^{{\frac{1}{{3}}}}=-{\left({2}^{ -{{1}}}\right)}\)

\(\displaystyle={\left(-\frac{1}{{2}}\right)}\)

Thus, the value of the expression \(\displaystyle-{\left(\frac{1}{{8}}\right)}^{{\frac{1}{{3}}}}\ \text{is}\ \displaystyle={\left(-\frac{1}{{2}}\right)}.\)

Definition of rational exponents:

"For any rational exponent m/n in its lowest terms, where m and n are integers and \(\displaystyle{n}>{0}\ \text{we define,}\ \displaystyle{\left({a}\right)}^{{\frac{m}{{n}}}}={\sqrt[{{n}}]{{{\left({a}\right)}^{m}}}}={\left({\sqrt[{{n}}]{{{a}}}}\right)}^{m}.\)

Formula used:

Laws of exponents:

"To raise a power to a new power, multiply the exponents".

\(\displaystyle{\left({a}^{m}\right)}^{n}={a}^{{{m}{n}}}\)

Calculation:

The given expression is \(\displaystyle{27}^{{\frac{1}{{3}}}}.\)

Use the above mentioned definition and simplify the expression as shown below.

\(\displaystyle{27}^{{\frac{1}{{3}}}}={\left({3}\times{3}\times{3}\right)}^{{\frac{1}{{3}}}}\)

\(\displaystyle={\left({3}^{3}\right)}^{{\frac{1}{{3}}}}\)

\(\displaystyle={3}^{{{3}\times\frac{1}{{3}}}}\)

\(= 3\)

Thus, the value of the expression \(\displaystyle{27}^{{\frac{1}{{3}}}}\) is 3.

(b) Use the above definition and formula mentioned in sub part (a) and simplify the expression as shown below.

\(\displaystyle{\left(-{8}\right)}^{{\frac{1}{{3}}}}={\left({\left(-{2}\right)}\times{\left(-{2}\right)}\times{\left(-{2}\right)}\right)}^{{\frac{1}{{3}}}}\)

\(\displaystyle={\left(-{2}\right)}^{{{3}\times\frac{1}{{3}}}}\)

\(\displaystyle={\left(-{2}\right)}^{{{3}\times\frac{1}{{3}}}}\)

\(=\ - 2\)

Thus, the value of the expression \(\displaystyle{\left(-{8}\right)}^{{\frac{1}{{3}}}}\) is (-2).

(c) Use the above definition and formula mentioned in sub part (a) and simplify the expression as shown below.

\(\displaystyle-{\left(\frac{1}{{8}}\right)}^{{\frac{1}{{3}}}}=-{\left(\frac{1}{{2}^{3}}\right)}^{{\frac{1}{{3}}}}\)

\(\displaystyle=-{\left({2}^{ -{{3}}}\right)}^{{\frac{1}{{3}}}}\)

\(\displaystyle=-{\left({2}^{{-{3}\times\frac{1}{{3}}}}\right)}\)

On further simplifications, the following is obtained.

\(\displaystyle-{\left(\frac{1}{{8}}\right)}^{{\frac{1}{{3}}}}=-{\left({2}^{ -{{1}}}\right)}\)

\(\displaystyle={\left(-\frac{1}{{2}}\right)}\)

Thus, the value of the expression \(\displaystyle-{\left(\frac{1}{{8}}\right)}^{{\frac{1}{{3}}}}\ \text{is}\ \displaystyle={\left(-\frac{1}{{2}}\right)}.\)