Question

simplify using laws of exponents : a) 64^-1/3 x [64^1/3 - 64^2/3]​

Answer 1

$\red{\bf {Answer:}}$

$64^{-\frac{1}{3}}(64^{\frac{1}{3}}-64^{\frac{2}{3}}$

$\green{\bf {Given}}$

$64^{-\frac{1}{3}}(64^{\frac{1}{3}}-64^{\frac{2}{3}}$

$\blue{\bf {Solution}}$

$=(4^3)^{-\frac{1}{3}}((4^3)^{\frac{1}{3}}-(4^3)^{\frac{2}{3}}$

$\purple{\bf {Applying \:exponent\:identity}}$

$(a^b)^x=a^{b\times x}$

$=(4)^{-\frac{3}{3}}((4)^{\frac{3}{3}}-(4)^{\frac{2\times 3}{3}}$

$=(4)^{-1}((4)^{1}-(4)^{2}$

$=(4)^{-1}(4-16)$

$=\frac{1}{4}\times(-12)$

$\red{\bf {Therefore}}$

$64^{-\frac{1}{3}}(64^{\frac{1}{3}}-64^{\frac{2}{3)}}$

More info :-

$\begin{gathered}\boxed{\begin{array}{| c |}\qquad\tt\large\textsf{ ! Formulas ! }\\\\\\\qquad\tt{:}\longrightarrow\large\textsf{ ( a + b )² = a² + 2ab + b² }\\\\\\\qquad\tt{:}\longrightarrow\large\textsf{ ( a - b )² = a² - 2ab + b² }\\\\\\\qquad\tt{:}\longrightarrow\large\textsf{ a² - b² = ( a + b ) ( a - b ) }\\\\\\\qquad\tt{:}\longrightarrow\large\textsf{ ( a + b )³ = a³ + 3a²b + 3ab² }\\\\\\\qquad\tt{:}\longrightarrow\large\textsf{ ( a - b )³ = a³ - 3a²b + 3ab² - b³ }\\\\\\\qquad\tt{:}\longrightarrow\large\textsf{ a³ - b³ = ( a - b ) ( a² + ab + b² ) }\end{array}}\end{gathered}$

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Answer 2

Step-by-step explanation:

$\bf \underline{Given-}$

$\sf{ {64}^{ - \frac{1}{3} } \bigg [ {64}^{ \frac{1}{3} } - {64}^{ \frac{2}{3} } \bigg ] }$

$\bf \underline{To \: find-}$

$\textsf{Simplify the given fractional expression and find their value.}$

$\bf \underline{Solution-}$

$\textsf{Given fractional expression,}$

$\sf{ {64}^{ - \frac{1}{3} } \bigg [ {64}^{ \frac{1}{3} } - {64}^{ \frac{2}{3} } \bigg ] }$

$\sf{ \Rightarrow \:( {4}^{3} {)}^{ - \frac{1}{3} } \bigg[( {4}^{3} {)}^{ \frac{1}{3} } - ( {4}^{3} {)}^{ \frac{2}{3} } \bigg] }$

$\sf{ \Rightarrow \: {4}^{ \cancel3 \times \big( - \frac{1}{ \cancel3} \big) } \bigg( {4}^{ \cancel3 \times \frac{1}{ \cancel3} } - {4}^{ \cancel3 \times \frac{2}{ \cancel3} } \bigg) }$

$\sf{ \Rightarrow \: {4}^{ - 1} (4 - {4}^{2}) }$

$\sf{ \Rightarrow \: \frac{1}{4} (4 - {4}^{2} )}$

$\sf{ \Rightarrow \: \frac{1}{4} (4 - 16) }$

$\sf{ \Rightarrow \: \frac{1}{ \cancel4} \times ( - \cancel{ {12}}^{3} )}$

$\sf{ \Rightarrow \: - 3}$

$\bf \underline{Answer-}$

$\bf{ \underline{Hence \: after \: simplifying \: we \: get \: the \: value \: of : }}$

$\sf \underline{\boxed{ \sf{ {64}^{ - \frac{1}{3} } \bigg [ {64}^{ \frac{1}{3} } - {64}^{ \frac{2}{3} } \bigg ] } \: is \: -3 } }$