 # Rational exponents evaluate each expression. (a) 27^(1/3) (b) (-8)^(1/3) (c) -(1/8)^(1/3) Khaleesi Herbert 2020-10-27 Answered

Rational exponents evaluate each expression.
(a) $$\displaystyle{27}^{{{1}\text{/}{3}}}$$
(b) $$\displaystyle{\left(-{8}\right)}^{{{1}\text{/}{3}}}$$
(c) $$\displaystyle-{\left(\frac{1}{{8}}\right)}^{{{1}\text{/}{3}}}$$

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