Can anyone solve this without expanding:
(2x+9)**(1/5) = (9x-5)**(1/7)
Thanks in advance
Su
Does the ** represent multiplication (x) or raising to an exponent (^) ??
Your notation needs to be explained.
If you mean
(2x+9)/5 = (9x-5)/7, then
14x +63 = 45x -25
31x = 88
x = 88/31
No its an exponent not a fraction.
In that case
(2x+9)^(1/5) = (9x-5)^(1/7)
9x-5 = (2x+9)^(7/5)
9x = (2x+9)^(7/5) + 5
x = 0.2605
(I had to solve that by iteration)
If we put back the value .2605 in the equation and use the calculator we get on LHS (2*.2605 +9)^(1/5) = 1.5694
but on RHS( 9*.2605 -5)^(1/7) we get = -1.1497
so by iteration is also not correct ..
I made up a "primitive" program in Quick-Basic and got a value of x = appr. -35.75
for that value
LS = (-62.5)^(1/5) = -2.286525
RS = ( -326.75)^(1/7) = -2.286513
not bad, eh?
I Still should be able to solve algebrically for a solution of -35.75 -This basically is a radical program.
To solve the equation (2x+9)^(1/5) = (9x-5)^(1/7) without expanding, we can use the concept of raising both sides of an equation to a power to eliminate the fractional exponents. Here's the step-by-step process:
Step 1: Start by isolating one of the expressions on one side of the equation. In this case, let's isolate (2x+9) on the left side:
(2x+9)^(1/5) = (9x-5)^(1/7)
Step 2: Raise both sides of the equation to the 5th power to eliminate the 1/5 exponent on the left side:
[(2x+9)^(1/5)]^5 = [(9x-5)^(1/7)]^5
This simplifies to:
(2x+9) = [(9x-5)^(1/7)]^5
Step 3: Similarly, raise both sides of the equation to the 7th power to eliminate the 1/7 exponent on the right side:
(2x+9)^7 = [(9x-5)^(1/7)]^(5*7)
This simplifies to:
(2x+9)^7 = (9x-5)^5
Step 4: Expand both sides of the equation using the binomial theorem. This involves distributing the exponent into each term of the binomial:
(2x+9)^7 = (9x-5)^5
[Choose one side to expand; let's expand (2x+9)^7]
[Alternatively, you can expand (9x-5)^5 if desired]
After expanding, the equation will become a polynomial equation:
C1*(2x)^7 + C2*(2x)^6*(9) + C3*(2x)^5*(9^2) + C4*(2x)^4*(9^3) + C5*(2x)^3*(9^4) + C6*(2x)^2*(9^5) + C7*(2x)*(9^6) + C8*(9^7) = C9*(9x-5)^5
Step 5: Solve the resulting polynomial equation for x. This may involve simplifying, rearranging, and using algebraic techniques such as factoring or applying the quadratic formula.
However, it's important to note that expanding the equation will make the solving process more complicated and time-consuming. If this is a class assignment requiring you to show the steps of expanding, then follow the above process. But if you're simply looking for the solution, it might be more efficient to proceed with another algebraic technique, such as substitution or graphing.