Question

# To calculate: The expression displaystylefrac{{2.7}^{{-{11}text{/}{12}}}}{{2.7}^{{-{1}text{/}{6}}}} with positive exponent.

To calculate:
The expression $$\displaystyle\frac{{2.7}^{{-{11}\text{/}{12}}}}{{2.7}^{{-{1}\text{/}{6}}}}$$ with positive exponent.

2020-12-14
Formula used:
Law of exponent: For any real number a and b and rational exponents m and n for which $$a^{m}\ \text{and}\ a^{n}$$ are defined, when deviding, exponents are subtracted for same base, that is,
$$\displaystyle\frac{{a}^{m}}{{a}^{n}}={a}^{{{m}-{n}}}$$
Negative rational exponents:
For any non-zero number a for which $$\displaystyle{a}^{{{m}\text{/}{n}}}\ \text{exist, the expression}\ \displaystyle{a}^{{-{m}\text{/}{n}}}\ \text{means}\ \displaystyle\frac{1}{{a}^{{{m}\text{/}{n}}}}.$$
Calculation:
Consider the given expression $$\displaystyle\frac{{2.7}^{{-{11}\text{/}{12}}}}{{2.7}^{{-{1}\text{/}{6}}}}.$$
As the above expression have same base. So, use exponent law $$\displaystyle\frac{{a}^{m}}{{a}^{n}}={a}^{{{m}-{n}}},$$
$$\displaystyle\frac{{2.7}^{{-{11}\text{/}{12}}}}{{2.7}^{{-{1}\text{/}{6}}}}={2.7}^{{-{11}\text{/}{12}-{\left(-{1}\text{/}{6}\right)}}}$$
As, the denominator of exponent is same, so,
$$\displaystyle{2.7}^{{-{11}\text{/}{12}-{\left(-{1}\text{/}{6}\right)}={2.7}^{{-{11}\text{/}{12}-{\left(-{2}\text{/}{6}\right)}}}}}$$
$$\displaystyle={2.7}^{{-{9}\text{/}{12}}}$$
$$\displaystyle={2.7}^{{-{3}\text{/}{4}}}$$
So, by reciprocating the base value of negative exponent,
$$\displaystyle{2.7}^{{-{3}\text{/}{4}}}=\frac{1}{{{2.7}^{{{3}\text{/}{4}}}}}$$
Thus, the value of expression $$\displaystyle\frac{{2.7}^{{-{11}\text{/}{12}}}}{{2.7}^{{-{1}\text{/}{6}}}}{i}{s}\frac{1}{{{2.7}^{{{3}\text{/}{4}}}.}}$$