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Answer :
Let's solve the equation step-by-step:
We start with the given equation:
[tex]\[ 125^{2x} \cdot 625 = 125^{3x} \][/tex]
The first step is to express 625 as a power of 125. We know that:
[tex]\[ 625 = 125^2 \][/tex]
Substituting this into the equation, we have:
[tex]\[ 125^{2x} \cdot 125^2 = 125^{3x} \][/tex]
Now, we can use the property of exponents:
[tex]\[ a^m \cdot a^n = a^{m+n} \][/tex]
Applying this property to the left side of our equation gives:
[tex]\[ 125^{2x + 2} = 125^{3x} \][/tex]
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[ 2x + 2 = 3x \][/tex]
Next, we solve for [tex]\( x \)[/tex]. Subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[ 2 = 3x - 2x \][/tex]
This simplifies to:
[tex]\[ 2 = x \][/tex]
Therefore, the solution to the equation is [tex]\( x = 2 \)[/tex].
We start with the given equation:
[tex]\[ 125^{2x} \cdot 625 = 125^{3x} \][/tex]
The first step is to express 625 as a power of 125. We know that:
[tex]\[ 625 = 125^2 \][/tex]
Substituting this into the equation, we have:
[tex]\[ 125^{2x} \cdot 125^2 = 125^{3x} \][/tex]
Now, we can use the property of exponents:
[tex]\[ a^m \cdot a^n = a^{m+n} \][/tex]
Applying this property to the left side of our equation gives:
[tex]\[ 125^{2x + 2} = 125^{3x} \][/tex]
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[ 2x + 2 = 3x \][/tex]
Next, we solve for [tex]\( x \)[/tex]. Subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[ 2 = 3x - 2x \][/tex]
This simplifies to:
[tex]\[ 2 = x \][/tex]
Therefore, the solution to the equation is [tex]\( x = 2 \)[/tex].
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