Math

Question Expand 2a3b4(a1b3a2b2)2 a^{3} b^{4}\left(a^{-1} b-3 a^{-2} b^{2}\right) using the distributive property.

Studdy Solution

STEP 1

Assumptions1. We are given the expression a3b4(a1b3ab) a^{3} b^{4}\left(a^{-1} b-3 a^{-} b^{}\right). We need to expand this expression using the distributive property

STEP 2

The distributive property states that for any real numbers a, b, and c, the equation a * (b + c) = a * b + a * c holds. We can apply this property to the given expression.

STEP 3

First, distribute 2a3b2 a^{3} b^{} to a1ba^{-1} b.
2a3ba1b2 a^{3} b^{} * a^{-1} b

STEP 4

Next, distribute 2a3b42 a^{3} b^{4} to 3a2b2-3 a^{-2} b^{2}.
2a3b43a2b22 a^{3} b^{4} * -3 a^{-2} b^{2}

STEP 5

Now we simplify the expressions. When multiplying terms with the same base, we add the exponents.
2a3b4a1b=2a3+(1)b4+1=2a2b52 a^{3} b^{4} * a^{-1} b =2 a^{3+(-1)} b^{4+1} =2 a^{2} b^{5}

STEP 6

Similarly, simplify the second expression.
2a3b43a2b2=6a3+(2)b4+2=6a1b62 a^{3} b^{4} * -3 a^{-2} b^{2} = -6 a^{3+(-2)} b^{4+2} = -6 a^{1} b^{6}

STEP 7

Finally, we write the expanded form of the original expression.
2a3b4(a1b3a2b2)=2a2b56a1b62 a^{3} b^{4}\left(a^{-1} b-3 a^{-2} b^{2}\right) =2 a^{2} b^{5} -6 a^{1} b^{6}So, the expanded form of the given expression is 2a2b56a1b62 a^{2} b^{5} -6 a^{1} b^{6}.

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