Evaluate the expression E_4= ( ( b^ 3 b^ -2 )^ -1 , ,(b^ -4 )^ -2 )

MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

Evaluate the expression

E_4=\left(\left(\dfrac{b^{3}}{b^{-2}}\right)^{-1}\,\cdot\,(b^{-4})^{-2}\right),\qquad b\neq0,

and answer the following questions. (i) Simplify

\left(\dfrac{b^{3}}{b^{-2}}\right)^{-1}

to a single power of

b

. (ii) Simplify

(b^{-4})^{-2}

to a single power of

b

. (iii) Hence, simplify

E_4

completely and write it in the form

b^{k}

, stating the value of

k

. (iv) Verify your result in part (iii) by substituting

b = 5

and calculating both the original and simplified expressions.