What is a^{-3}b^2(ba^3+b^{-1}a^2+b^{-2}a) ?

The solution to a^{-3}b^2(ba^3+b^{-1}a^2+b^{-2}a) is (a^2b^3+ab+1)/(a^2)

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simplify a^{-3}b^2(ba^3+b^{-1}a^2+b^{-2}a)

simplify

a^{- 3} b^{2} (b a^{3} + b^{- 1} a^{2} + b^{- 2} a)

\frac{a ^{2} b ^{3} + ab + 1}{a ^{2}}

Solution steps

a^{- 3} b^{2} (b a^{3} + b^{- 1} a^{2} + b^{- 2} a)

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

a^{- 3} = \frac{1}{a ^{3}}

= \frac{1}{a ^{3}} b^{2} (b a^{3} + b^{- 1} a^{2} + b^{- 2} a)

Simplify

b a^{3} + b^{- 1} a^{2} + b^{- 2} a : b a^{3} + \frac{a ^{2}}{b} + \frac{a}{b ^{2}}

= \frac{1}{a ^{3}} b^{2} (b a^{3} + \frac{a ^{2}}{b} + \frac{a}{b ^{2}})

Apply the fraction rule:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot b ^{2} ( b a ^{3} + \frac{a ^{2}}{b} + \frac{a}{b ^{2}} )}{a ^{3}}

Simplify

1 \cdot b^{2} (b a^{3} + \frac{a ^{2}}{b} + \frac{a}{b ^{2}}) : a^{3} b^{3} + a^{2} b + a

= \frac{a ^{3} b ^{3} + a ^{2} b + a}{a ^{3}}

Factor

a^{3} b^{3} + a^{2} b + a : a (a^{2} b^{3} + ab + 1)

= \frac{a ( a ^{2} b ^{3} + ab + 1 )}{a ^{3}}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

a^{3} = a a^{2}

= \frac{a ( a ^{2} b ^{3} + ab + 1 )}{a a ^{2}}

Cancel the common factor:

a

= \frac{a ^{2} b ^{3} + ab + 1}{a ^{2}}

a^{3} b^{3}

(0.5 m^{6} p^{5}) (0.2 m^{2} n)

6 5^{3}

x^{x}

n e^{- n^{2}}

What is a^{-3}b^2(ba^3+b^{-1}a^2+b^{-2}a) ?
The solution to a^{-3}b^2(ba^3+b^{-1}a^2+b^{-2}a) is (a^2b^3+ab+1)/(a^2)