Math

QuestionSimplifica la expresión A=2m+62(2m+1)2m+3(2m1)A=\frac{2^{m+6}-2\left(2^{m+1}\right)}{2^{m}+3\left(2^{m-1}\right)}.

Studdy Solution

STEP 1

Assumptions1. The given expression is A=m+6(m+1)m+3(m1)A=\frac{^{m+6}-\left(^{m+1}\right)}{^{m}+3\left(^{m-1}\right)}. We are asked to simplify this expression.

STEP 2

First, we can simplify the numerator by factoring out the common term 2m+12^{m+1}.
A=2m+1(252)2m+(2m1)A=\frac{2^{m+1}(2^5-2)}{2^{m}+\left(2^{m-1}\right)}

STEP 3

Now, we can simplify the term inside the parentheses in the numerator.
A=2m+1(322)2m+3(2m1)A=\frac{2^{m+1}(32-2)}{2^{m}+3\left(2^{m-1}\right)}

STEP 4

Calculate the value inside the parentheses in the numerator.
A=2m+1(30)2m+3(2m1)A=\frac{2^{m+1}(30)}{2^{m}+3\left(2^{m-1}\right)}

STEP 5

Now, we can simplify the denominator by factoring out the common term 2m12^{m-1}.
A=2m+1(30)2m1(2+3)A=\frac{2^{m+1}(30)}{2^{m-1}(2+3)}

STEP 6

Calculate the value inside the parentheses in the denominator.
A=2m+1(30)2m1(5)A=\frac{2^{m+1}(30)}{2^{m-1}(5)}

STEP 7

Now, we can simplify the expression by canceling out the common factors.
A=2m+1+1(30)5A=\frac{2^{m+1+1}(30)}{5}

STEP 8

implify the exponent in the numerator.
A=2m+2(30)5A=\frac{2^{m+2}(30)}{5}

STEP 9

Now, we can simplify the expression by dividing30 by5.
A=2m+2(6)A=2^{m+2}(6)

STEP 10

Finally, we can simplify the expression by multiplying6 by 2m+22^{m+2}.
A=62m+2A=6\cdot2^{m+2}So, the simplified form of the given expression is A=62m+2A=6\cdot2^{m+2}.

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