Question

`if \: ( (3)/(4) )^(6) * ( (16)/(9) )^(5) = ( (4)/(3) )^(x + 2) . \: find \: x`

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Answer 1

2

Explanation:

`((3)/(4))^6 * ((16)/(9))^5=((4)/(3))^(x+2)`

`(3^6)/(4^6) \cdot (16^5)/(9^5)=(4^(x+2))/(3^(x+2))`

`(3^6)/(4^6) \cdot ((4^2)^5)/((3^2)^5)=(4^(x+2))/(3^(x+2))`

`(3^6)/(4^6) \cdot (4^(10))/(3^(10))=(4^(x+2))/(3^(x+2))`

`(3^6)/(3^(10)) \cdot (4^(10))/(4^6)=(4^(x+2))/(3^(x+2))`

`3^(-4) \cdot 4^(4)=4^(x+2)3^(-(x+2))`

This implies

x+2=4

and

-(x+2)=-4.

x+2=4 implies x=2 since subtract 2 on both sides gives us x=2.

Solving -(x+2)=-4 should give us the same value.

Multiply both sides by -1:

x+2=4

It is the same equation as the other.

You will get x=2 either way.

Let's check:

`((3)/(4))^6 * ((16)/(9))^5=((4)/(3))^(2+2)`

`((3)/(4))^6 * ((16)/(9))^5=((4)/(3))^(4)`

Put both sides into your calculator and see if you get the same thing on both sides:

Left hand side gives 256/81.

Right hand side gives 256/81.

Both side are indeed the same for x=2.