Master the Challenge: Solving 5^x + 25^x = 125^x

Exponential equations like 5^x + 25^x = 125^xtest your understanding of powers and mathematical logic. These equations often appear in algebra exams and competitive tests, making them essential for students and math enthusiasts alike. In this blog, we’ll simplify and solve this equation step by step, unlocking the secrets of exponential relationships.

Step-by-Step Solution

The equation 5^x + 25^x = 125^xinvolves powers of 5. By rewriting each term using the base 55, we can simplify the problem.

1.     Rewrite Each Term with Base 5:

o    Recall that 25=5^2 and 125= 5^3.

o    Replace 25^x with  (5^2)^x = 5^{2x} and 125^x with (5^3)^x = 5^{3x}.

The equation becomes:

5^x+5^{2x} = 5^{3x}

2.     Factorize the Left-Hand Side:

Take 5^x as a common factor:

5^x(1+5^x)= 5^{3x}

3.     Divide Both Sides by 5^x:

Since 5^x is not equal to 0, divide through by 5^x:

1+5^x = 5^{2x}

4.     Set y= 5^x:

To simplify further, let y= 5^x. The equation becomes:

1+y= y^2

5.     Rearrange into a Quadratic Equation:

y^2 - y - 1 = 0

6.     Solve the Quadratic Equation:

Use the quadratic formula we get :

Y = {1+ and minus underroot 5}/2

7.     Find x:

Recall y=5^x, so:

5^x={1+ and minus underroot 5}/2  

Solve for xx using logarithms:

x=log{1+ and minus underroot 5}/2

Final Answer

The solution to 5^x + 25^x = 125^x is:

x=log( base 5) {1+ and minus underroot 5}/2  

Why These Problems Matter

Exponential equations like 5^x + 25^x = 125^x enhance your understanding of algebra and prepare you for advanced topics like logarithms and calculus. They’re widely used in physics, finance, and computer science, making them a valuable skill for both academic and real-world problem-solving.

 

Conclusion

Solving 5x+25x=125x5^x + 25^x = 125^x involves simplifying exponents and applying quadratic techniques. Practice problems like these to sharpen your algebra skills and unlock the full potential of exponential equations.

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