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hello and welcome how to solve this exponential equation 27 to^ x + 64 to^ x over 4^ x + 36 x = 193 144 find the values of X let us get started let us Begin by transforming this equation on the left hand side by expressing these elements in index form and we want to apply this rule in changing the form that a to power P of this to power C is the same as is the same as a to power CP and this one can be written as a to power C all of this to power B so this is the rule we want to apply throughout the transformation of the left hand side this is a very powerful rule which is going to transform the equation above now next we have 27 is the same as 3 cubed so so we have 3 cubed of this power X Plus 64 is the same as 4 to^ 3 of this power x over 48 is the same as 3 * 16 the of this to^ X plus this one 36 is the same as 4 * 9 so we have 4 * 9 of this to^ x now the all of this equals to 193 over 144 16 is the same as 4 S and 9 is the same as 3 squ now next this equation going to be written us 3 to^ 3 of this X plus 4^ 3 of this to^ X over 3 * 4 S of this x + 4 * 3 [Music] 2 the of this ra to^ x equal to 193 over 144 now let us apply this rule in changing the forms of this now we have 3^ x^ 3 because you can see the way the equation will change from here up to here so we have now next we have by by use of the relationship by use of this power rule we have 3^ X cubed plus 4^ X cubed over here we have 3^ x * 4 power x² plus we have 4^ x * 3^ x 2 all of this equals to 193 over 144 now let a = to 3^ X and P = to 4 X Now by substitution into the equation we have a cub + P cubed over a p 2 + P a² = to 193 over 144 now let us factorize this equation here let us use this relationship that M Cub + n Cub is the same as m + n into M 2us MN + n² we want to apply this on the numerator now next we have a + P into a 2 - a p + p 2 all of this over in the denominator a is common take it out so we have a into so we take a out we remain with B plus here we have taken a out we remain with a of this equals to 193 over 144 as you can see the order the matter here so this one cancels with this now next we have we have a 2 minus a p + p² of this a p = 193 144 now let us divide the num by denominator individually so we have next we have a 2 a pus a p a p + p² a p = to 193 over 144 by simplifying as you can see this one cancels of go to one times and here you see X cancels reduces this by power one and the B reduces this power by one now next we have a / P - 1 + P / a equals to 193 over 144 let us add one on both sides so that now next we have a / p + P / a = 193 144 + one I have added one on both sides so that we bring like terms together so here we have 193 144 + 1 is so we have 193 + 144 the of this 144 so this one gives us 3 37 over 144 now next the equation becomes a / p+ P / a = to 337 / 144 now let let M be equals to a/ p and 1 / m equal to P / a Now by substitution into this equation we have m + 1 / M = to 337 over 144 by cross multiplication to eliminate the fractions let us multiply both sides of the by the LM which is 144 M so multiply both sides multiply both sides of the equation by 44 144 M by 144 M so we have we have 144 m^ 2+ 144 equal to 337 M by writing the standard form of quadratic equation so this one becomes 144 m^ 2 - 337 m + 144 = 0 now let us use factoring method in which the product here is 144 s because a * C 144 * 144 you get this sum = to - 3 37 so the factors that give us this are -256 * 81 now let us use this factors in this equation so next we have 144 m 2 - 256 M - 81 m + 144 = 0 let us factorize the first two and the last two times so in the first two terms 16 m is common so we take it out so we have 16 M into 9 M - 6 16 minus the next two terms 9 9 is common so we have 9 into 9 M - 16 = 0 now as you can see this equation we have 16 M into 9 M - 16 - 9 into 9 M - 16 = to 0 now let us pull out 9 M minus let us pull out 9 M - 16 so we have 9 M - 16 into 16 M - 9 = to 0 by the application of the Z product we have 9 is 9 M - 16 = 0 or 16 M - 9 = 0 so this one gives us m = 16 / 9 or m = 9 / 16 part a/ P = to M which means that a/ P = to 16 / 9 or a / P = 9 / 16 recall a equals to 3^ X and P = to 4^ X so it implies that by putting this parameters here we have 3^ x / 4^ x = 16 / 9 or 3 to^ x/ 4^ x = 9 / 16 now if we solve this to we have by use of this relationship that a^ n p^ n is the same as a p of this n now by the application of this rule then we have 3/ 4 of this to^ x = 16 / 9 or 3 3 over 4 of this to^ x = 9 / 16 now let us express the right hand side of this equations in index form so so we have 3 / 4 this x = to this one is as 4 2/ 3 2 or 3 / 4^ X is the same as 3 S over 4 S now by the application of this rule once again on the S now we have 3 / 4 this x = 4 3 this power 2 3 4 the of this x = 3 4 this to power two now as you can see this one is pretty straightforward here the PES are the same it implies that X = two as one of the solutions of the equation and here let us apply this rule that on this one apply this that that m/ n l to C is the same as n/ M the of this ra to C so this a very powerful rule we going to apply here so this one becomes now the right hand side becomes so we have 3 4 of this power xal to 3 4 now by the of the root this exponent changes to negative so it implies that X here = -2 so it means that here x = to -2 now we have two solutions X = 2 and X2 that satisfy our equation and that is it thank you for watching subscribe to my channel and turn on the notification Bell to get new updates when I upload new videos don't forget to smash like button share and comment below to make this video many people