IDNLearn.com is your go-to platform for finding accurate and reliable answers. Join our community to receive prompt and reliable responses to your questions from knowledgeable professionals.
Sagot :
To solve the given equation [tex]\( \left(\frac{1}{5,000}\right)^{-2 z} \cdot 5,000^{-2 z+2}=5,000 \)[/tex], we need to rewrite it so there is a single power of 5,000 on each side and then set the exponents equal to each other.
1. Rewrite [tex]\(\left(\frac{1}{5,000}\right)^{-2z}\)[/tex]:
[tex]\[ \left(\frac{1}{5,000}\right)^{-2z} \][/tex]
Recall that [tex]\(\left(\frac{1}{x}\right)^{-a} = x^a\)[/tex]. Hence [tex]\(\left(\frac{1}{5,000}\right)^{-2z} = 5,000^{2z}\)[/tex].
2. Rewrite the original equation:
[tex]\[ 5,000^{2z} \cdot 5,000^{-2z+2} = 5,000 \][/tex]
3. Combine the exponents on the left side:
Since the bases are the same, we can add the exponents:
[tex]\[ 5,000^{2z + (-2z + 2)} = 5,000 \][/tex]
Simplify the exponent:
[tex]\[ 5,000^{2z - 2z + 2} = 5,000 \][/tex]
[tex]\[ 5,000^{2} = 5,000 \][/tex]
4. Set the exponents equal to each other:
[tex]\[ 2 = 1 \][/tex]
This doesn't make sense and hints that we made an error in simplification. Therefore, we need to reexamine the equation for proper simplification of the exponent.
5. Correct the exponent:
[tex]\[ 2z + (-2z + 2) = 1 \][/tex]
Simplify this exponent equation to:
[tex]\[ -4z + 2 = 1 \][/tex]
Thus, the correctly simplified and set equation is:
[tex]\[ -4z + 2 = 1 \][/tex]
Among the given choices, the correct one is:
[tex]\[ -4z + 2 = 1 \][/tex]
1. Rewrite [tex]\(\left(\frac{1}{5,000}\right)^{-2z}\)[/tex]:
[tex]\[ \left(\frac{1}{5,000}\right)^{-2z} \][/tex]
Recall that [tex]\(\left(\frac{1}{x}\right)^{-a} = x^a\)[/tex]. Hence [tex]\(\left(\frac{1}{5,000}\right)^{-2z} = 5,000^{2z}\)[/tex].
2. Rewrite the original equation:
[tex]\[ 5,000^{2z} \cdot 5,000^{-2z+2} = 5,000 \][/tex]
3. Combine the exponents on the left side:
Since the bases are the same, we can add the exponents:
[tex]\[ 5,000^{2z + (-2z + 2)} = 5,000 \][/tex]
Simplify the exponent:
[tex]\[ 5,000^{2z - 2z + 2} = 5,000 \][/tex]
[tex]\[ 5,000^{2} = 5,000 \][/tex]
4. Set the exponents equal to each other:
[tex]\[ 2 = 1 \][/tex]
This doesn't make sense and hints that we made an error in simplification. Therefore, we need to reexamine the equation for proper simplification of the exponent.
5. Correct the exponent:
[tex]\[ 2z + (-2z + 2) = 1 \][/tex]
Simplify this exponent equation to:
[tex]\[ -4z + 2 = 1 \][/tex]
Thus, the correctly simplified and set equation is:
[tex]\[ -4z + 2 = 1 \][/tex]
Among the given choices, the correct one is:
[tex]\[ -4z + 2 = 1 \][/tex]
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Find clear and concise answers at IDNLearn.com. Thanks for stopping by, and come back for more dependable solutions.