To simplify the expression (-27)^{\frac{1}{3}}

To simplify the expression $$\displaystyle{\left(-{27}\right)}^{{{\frac{{{1}}}{{{3}}}}}}$$

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Step 1
According to the power rule $$\displaystyle{\left({b}^{{{n}}}\right)}{m}={b}^{{{n}}}\cdot{m}$$
Given expression is $$\displaystyle{\left(-{27}\right)}^{{{\frac{{{1}}}{{{3}}}}}}$$
This can be written as $$\displaystyle{\left(-{27}\right)}^{{{\frac{{{1}}}{{{3}}}}}}={\left\lbrace{\left(-{3}\right)}\times{\left(-{3}\right)}\times{\left(-{3}\right)}\right\rbrace}^{{{\frac{{{1}}}{{{3}}}}}}$$
$$\displaystyle{\left(-{27}\right)}^{{{\frac{{{1}}}{{{3}}}}}}={\left\lbrace{\left(-{3}\right)}^{{{3}}}\right\rbrace}^{{{\frac{{{1}}}{{{3}}}}}}$$
Applying the power rule,
$$\displaystyle{\left(-{27}\right)}^{{{\frac{{{1}}}{{{3}}}}}}={\left\lbrace{\left(-{3}\right)}\right\rbrace}^{{{3}\times{\frac{{{1}}}{{{3}}}}}}$$
$$\displaystyle{\left(-{27}\right)}^{{{\frac{{{1}}}{{{3}}}}}}={\left(-{3}\right)}$$