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anemone
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Assume that $a,\,b,\,c$ and $d$ are positive integers such that $a^5=b^4,\,c^3=d^2$ and $c-a=19$.
Determine $d-b$.
Determine $d-b$.
anemone said:Assume that $a,\,b,\,c$ and $d$ are positive integers such that $a^5=b^4,\,c^3=d^2$ and $c-a=19$.
Determine $d-b$.
To solve for $d-b$, we need to isolate the variable on one side of the equation. In this case, we can use the given equations to substitute values for $a$, $b$, and $c$ in terms of $d$. Then, we can solve for $d$ and plug the value back into the equation to find $b$. Finally, we can subtract $b$ from $d$ to get the value of $d-b$.
First, we need to rearrange the equations to isolate $d$ on one side. We can use the fact that $a^5=b^4$ to substitute $a$ in terms of $b$ in the second equation. Then, we can substitute $d$ in terms of $c$ in the first equation. Next, we can use the third equation to substitute $a$ in terms of $c$. Finally, we can solve for $d$ and plug the value back into the equation to find $b$. Subtracting $b$ from $d$ will give us the value of $d-b$.
There may be multiple methods to solve this equation, but the given steps are a systematic approach to finding the value of $d-b$. It may be possible to simplify the equations or use different substitutions, but the end result will still be the same.
Sure, let's say $a=2$, $b=4$, $c=27$. We can substitute these values into the equations to get $2^5=4^4$, $27^3=d^2$, and $27-2=19$. From the first equation, we know that $a^5=b^4$ and $2^5=32$, so $b^4=32$ and $b=2$. Substituting this into the second equation, we get $27^3=d^2$ and $d=27$. Finally, we can subtract $2$ from $27$ to get the value of $d-b$, which is $25$.
There may be certain restrictions or limitations depending on the given values in the equations. For example, if $a$ and $b$ are both negative, then the equation $a^5=b^4$ may not hold true. Additionally, if the given values do not result in a real solution for $d$, then the equation may not have a solution for $d-b$.