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].
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].
