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Answer :
Sure, let's solve this step-by-step:
We are given the equation:
[tex]\[ 5^{2n} \times 25^{2n-1} = 625. \][/tex]
Step 1: Rewrite the terms with the same base
The number 25 can be rewritten as a power of 5:
[tex]\[ 25 = 5^2. \][/tex]
Substitute this back into the equation:
[tex]\[ 5^{2n} \times (5^2)^{2n-1} = 625. \][/tex]
Step 2: Simplify the expression
Using the rules of exponents, [tex]\( (a^m)^n = a^{m \times n} \)[/tex], we can rewrite [tex]\( (5^2)^{2n-1} \)[/tex] as follows:
[tex]\[ (5^2)^{2n-1} = 5^{2 \times (2n-1)} = 5^{4n-2}. \][/tex]
Now the equation becomes:
[tex]\[ 5^{2n} \times 5^{4n-2} = 625. \][/tex]
Step 3: Combine the exponents
When multiplying powers with the same base, we add the exponents:
[tex]\[ 5^{2n + 4n - 2} = 625. \][/tex]
So, we have:
[tex]\[ 5^{6n - 2} = 625. \][/tex]
Step 4: Express 625 as a power of 5
Recognize that 625 is also a power of 5:
[tex]\[ 625 = 5^4. \][/tex]
Step 5: Equate the exponents
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[ 6n - 2 = 4. \][/tex]
Step 6: Solve for [tex]\( n \)[/tex]
Add 2 to both sides of the equation:
[tex]\[ 6n = 6. \][/tex]
Now, divide by 6 to solve for [tex]\( n \)[/tex]:
[tex]\[ n = 1. \][/tex]
Thus, the solution is [tex]\( n = 1 \)[/tex].
We are given the equation:
[tex]\[ 5^{2n} \times 25^{2n-1} = 625. \][/tex]
Step 1: Rewrite the terms with the same base
The number 25 can be rewritten as a power of 5:
[tex]\[ 25 = 5^2. \][/tex]
Substitute this back into the equation:
[tex]\[ 5^{2n} \times (5^2)^{2n-1} = 625. \][/tex]
Step 2: Simplify the expression
Using the rules of exponents, [tex]\( (a^m)^n = a^{m \times n} \)[/tex], we can rewrite [tex]\( (5^2)^{2n-1} \)[/tex] as follows:
[tex]\[ (5^2)^{2n-1} = 5^{2 \times (2n-1)} = 5^{4n-2}. \][/tex]
Now the equation becomes:
[tex]\[ 5^{2n} \times 5^{4n-2} = 625. \][/tex]
Step 3: Combine the exponents
When multiplying powers with the same base, we add the exponents:
[tex]\[ 5^{2n + 4n - 2} = 625. \][/tex]
So, we have:
[tex]\[ 5^{6n - 2} = 625. \][/tex]
Step 4: Express 625 as a power of 5
Recognize that 625 is also a power of 5:
[tex]\[ 625 = 5^4. \][/tex]
Step 5: Equate the exponents
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[ 6n - 2 = 4. \][/tex]
Step 6: Solve for [tex]\( n \)[/tex]
Add 2 to both sides of the equation:
[tex]\[ 6n = 6. \][/tex]
Now, divide by 6 to solve for [tex]\( n \)[/tex]:
[tex]\[ n = 1. \][/tex]
Thus, the solution is [tex]\( n = 1 \)[/tex].
Thank you for reading the article Solve the following equation i tex 5 2n times 25 2n 1 625 tex. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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Rewritten by : Jeany
