Question
Solve for x \( 25^{2 x+1}=125^{3 x+4} \) \( x= \)
Ask by Ford Chen. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x = -2 \)
Solution
Solve the equation \( 25^{2x+1}=125^{3x+4} \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(25^{2x+1}=125^{3x+4}\)
- step1: Rewrite the expression:
\(5^{4x+2}=5^{9x+12}\)
- step2: Set the exponents equal:
\(4x+2=9x+12\)
- step3: Move the expression to the left side:
\(4x-9x=12-2\)
- step4: Add and subtract:
\(-5x=12-2\)
- step5: Add and subtract:
\(-5x=10\)
- step6: Change the signs:
\(5x=-10\)
- step7: Divide both sides:
\(\frac{5x}{5}=\frac{-10}{5}\)
- step8: Divide the numbers:
\(x=-2\)
The solution to the equation \(25^{2x+1}=125^{3x+4}\) is \(x=-2\).
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Simplify this solution Bonus Knowledge
To solve the equation \( 25^{2x+1} = 125^{3x+4} \), let's first express both sides in terms of powers of 5. We know that \( 25 = 5^2 \) and \( 125 = 5^3 \). Substituting these values, we get: \[ (5^2)^{2x+1} = (5^3)^{3x+4} \] This simplifies to: \[ 5^{4x + 2} = 5^{9x + 12} \] Since the bases are the same, we can set the exponents equal to each other: \[ 4x + 2 = 9x + 12 \] Now, let's solve for \( x \). First, we'll isolate \( x \) on one side: \[ 2 - 12 = 9x - 4x \] \[ -10 = 5x \] \[ x = -2 \] Thus, the solution is: \[ x = -2 \]
