Rational exponents evaluate each expression.

(a)

(b)

(c)

Khaleesi Herbert
2020-10-27
Answered

Rational exponents evaluate each expression.

(a)

(b)

(c)

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d2saint0

Answered 2020-10-28
Author has **89** answers

(a) Definition used:

Definition of rational exponents:

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

Formula used:

Laws of exponents:

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

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

Calculation:

The given expression is${27}^{\frac{1}{3}}.$

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

$27}^{\frac{1}{3}}={(3\times 3\times 3)}^{\frac{1}{3}$

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

$={3}^{3\times \frac{1}{3}}$

$=3$

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.

$(-8)}^{\frac{1}{3}}={((-2)\times (-2)\times (-2))}^{\frac{1}{3}$

$={(-2)}^{3\times \frac{1}{3}}$

$={(-2)}^{3\times \frac{1}{3}}$

$=\text{}-2$

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

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

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

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

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

On further simplifications, the following is obtained.

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

$=(-\frac{1}{2})$

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

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".

Calculation:

The given expression is

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

Thus, the value of the expression

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

Thus, the value of the expression

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

On further simplifications, the following is obtained.

Thus, the value of the expression

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