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Question -
Evaluate [tex]\rm( {\frac{ - 1}{27})}^{ - \frac{2}{3} } [/tex]
Answer -
[tex]\rm ({\dfrac{ - 1}{27}) }^{ - \frac{2}{3} } [/tex] [tex]\rm = (\dfrac{27}{1})^{ \frac{2}{3} } [/tex]
[tex]\rm = ({3}^{3})^\frac{2}{3} [/tex]
[tex]\rm = 3 ^{(\cancel3 \times \frac{2}{ \cancel3})} [/tex]
[tex]\rm = {3}^{2} [/tex]
[tex]\rm = 9[/tex]
Additional information -
[tex]\boxed{\sf\gray{ a^m \times a^n = a^{m+n}}}[/tex]
[tex]\boxed{\sf\gray {\frac{a^m}{a^n} = a^{m-n}}}[/tex]
[tex]\boxed{\sf\gray{ (a^{m)n} = a^{mn} }}[/tex]
[tex]\boxed{\sf\gray{ a^{-n} = \frac{1}{a^n}}}[/tex]
[tex]\boxed{\sf\gray{ a^{\frac{m}{n}} = \sqrt[n]{a^m}}}[/tex]
[tex]\boxed{\sf\gray{(\frac{a}{b})^m = \frac{a^m}{b^m}}}[/tex]
where -
[tex]\longrightarrow[/tex]a and b are positive real numbers.
[tex]\longrightarrow[/tex]m and n are rational numbers.
[tex]\rm\underline\red{Question}[/tex]
To find the value of or to Evaluate:-
[tex]\sf⟾({\frac{-1}{27})}^{-\frac{2}{3}}[/tex]
[tex]\sf\underline\blue{Solution:-}[/tex]
[tex]\sf⟾({\frac{27}{1})}^{\frac{2}{3}}[/tex]
[tex]\sf⟾({27)}^{\frac{2}{3}}[/tex]
[tex]\sf⟾{3}^{3}{×\frac{2}{3}}[/tex]
[tex]\sf⟾{3}^{2}[/tex]
[tex]\sf⟾{3×3}[/tex]
[tex]\sf⟾{9}[/tex]
Therefore,after evaluation of [tex]\bf({\frac{-1}{27})}^{-\frac{2}{3}}[/tex] the result is [tex]\sf{9}[/tex].
Some laws of Exponents.