(Prove that) : $\frac{5^x-5^{x-1}}{4 \times 5^{x-1}}=1$
Answer (1)
Rewrite 5 x − 1 as 5 5 x .
Substitute 5 5 x for 5 x − 1 in the equation: 4 × 5 5 x 5 x − 5 5 x = 1 .
Simplify the numerator by factoring out 5 x : 4 × 5 5 x 5 x ( 5 4 ) = 1 .
The equation simplifies to 1 = 1 , proving the original equation: 1 = 1 .
Explanation
Understanding the Problem We are given the equation 4 × 5 x − 1 5 x − 5 x − 1 = 1 and we want to prove that it is true.
Rewriting the Term First, let's rewrite 5 x − 1 as 5 5 x . This will help us simplify the expression.
Substitution Now, substitute 5 5 x for 5 x − 1 in the equation: 4 × 5 5 x 5 x − 5 5 x = 1
Simplifying the Numerator Next, simplify the numerator by factoring out 5 x : 4 × 5 5 x 5 x ( 1 − 5 1 ) = 1 4 × 5 5 x 5 x ( 5 4 ) = 1
Simplifying the Denominator Now, simplify the denominator: 5 4 × 5 x 5 x ( 5 4 ) = 1
Final Proof Now, we can see that the numerator and the denominator are the same, so the fraction simplifies to 1: 5 4 × 5 x 5 x ( 5 4 ) = 5 4 × 5 x 5 4 × 5 x = 1 Thus, the equation is proven.
Examples
Exponential equations like this appear in many real-world scenarios, such as calculating population growth, radioactive decay, and compound interest. For example, if you invest money with compound interest, the amount you have after a certain time can be modeled using an exponential equation. Understanding how to simplify and solve these equations helps you predict future values and make informed decisions.
