# 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) ${27}^{1\text{/}3}$
(b) ${\left(-8\right)}^{1\text{/}3}$
(c) $-{\left(\frac{1}{8}\right)}^{1\text{/}3}$

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d2saint0
(a) Definition used:
Definition of rational exponents:
"For any rational exponent m/n in its lowest terms, where m and n are integers and
Formula used:
Laws of exponents:
"To raise a power to a new power, multiply the exponents".
${\left({a}^{m}\right)}^{n}={a}^{mn}$
Calculation:
The given expression is ${27}^{\frac{1}{3}}.$
Use the above mentioned definition and simplify the expression as shown below.
${27}^{\frac{1}{3}}={\left(3×3×3\right)}^{\frac{1}{3}}$
$={\left({3}^{3}\right)}^{\frac{1}{3}}$
$={3}^{3×\frac{1}{3}}$
$=3$
Thus, the value of the expression ${27}^{\frac{1}{3}}$ is 3.
(b) Use the above definition and formula mentioned in sub part (a) and simplify the expression as shown below.
${\left(-8\right)}^{\frac{1}{3}}={\left(\left(-2\right)×\left(-2\right)×\left(-2\right)\right)}^{\frac{1}{3}}$
$={\left(-2\right)}^{3×\frac{1}{3}}$
$={\left(-2\right)}^{3×\frac{1}{3}}$

Thus, the value of the expression ${\left(-8\right)}^{\frac{1}{3}}$ is (-2).
(c) Use the above definition and formula mentioned in sub part (a) and simplify the expression as shown below.
$-{\left(\frac{1}{8}\right)}^{\frac{1}{3}}=-{\left(\frac{1}{{2}^{3}}\right)}^{\frac{1}{3}}$
$=-{\left({2}^{-3}\right)}^{\frac{1}{3}}$
$=-\left({2}^{-3×\frac{1}{3}}\right)$
On further simplifications, the following is obtained.
$-{\left(\frac{1}{8}\right)}^{\frac{1}{3}}=-\left({2}^{-1}\right)$
$=\left(-\frac{1}{2}\right)$
Thus, the value of the expression