College

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ Use the like bases property to solve the equation [tex]$625^n=3125$[/tex].

[tex]n = \square[/tex]

Answer :

To solve the equation [tex]\( 625^n = 3125 \)[/tex] using the like bases property, follow these steps:

1. Express Both Sides as Powers of a Common Base:
First, we need to see if both numbers can be expressed as powers of a common base. Notice that both 625 and 3125 are powers of 5.
- [tex]\( 625 = 5^4 \)[/tex]
- [tex]\( 3125 = 5^5 \)[/tex]

2. Rewrite the Equation:
Substitute these expressions into the original equation:
[tex]\[
(5^4)^n = 5^5
\][/tex]

3. Apply the Power of a Power Property:
According to the power of a power property, [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Using this, express the left side:
[tex]\[
5^{4n} = 5^5
\][/tex]

4. Use the Like Bases Property:
When the bases are the same, the exponents must be equal for the equation to hold:
[tex]\[
4n = 5
\][/tex]

5. Solve for [tex]\( n \)[/tex]:
To find [tex]\( n \)[/tex], divide both sides of the equation by 4:
[tex]\[
n = \frac{5}{4}
\][/tex]

So, the solution to the equation [tex]\( 625^n = 3125 \)[/tex] is [tex]\( n = 1.25 \)[/tex].