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Answer :
To factor the expression [tex]\(\frac{1}{25} - 16z^2\)[/tex], we can recognize it as a difference of squares. The difference of squares formula is:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
First, we need to identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] such that [tex]\(a^2 = \frac{1}{25}\)[/tex] and [tex]\(b^2 = 16z^2\)[/tex].
1. Notice that [tex]\(\frac{1}{25}\)[/tex] is a perfect square. It can be written as [tex]\((\frac{1}{5})^2\)[/tex]. So here, [tex]\(a = \frac{1}{5}\)[/tex].
2. Likewise, [tex]\(16z^2\)[/tex] is a perfect square as well. This can be expressed as [tex]\((4z)^2\)[/tex]. So here, [tex]\(b = 4z\)[/tex].
Now we can apply the difference of squares formula:
[tex]\[ \frac{1}{25} - 16z^2 = \left(\frac{1}{5}\right)^2 - (4z)^2 \][/tex]
This can be factored as:
[tex]\[ \left(\frac{1}{5} - 4z\right)\left(\frac{1}{5} + 4z\right) \][/tex]
This is the factored form of the expression. Therefore, the correct choice is:
A. [tex]\(\frac{1}{25}-16 z^2 = \left(\frac{1}{5} - 4z\right)\left(\frac{1}{5} + 4z\right)\)[/tex]
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
First, we need to identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] such that [tex]\(a^2 = \frac{1}{25}\)[/tex] and [tex]\(b^2 = 16z^2\)[/tex].
1. Notice that [tex]\(\frac{1}{25}\)[/tex] is a perfect square. It can be written as [tex]\((\frac{1}{5})^2\)[/tex]. So here, [tex]\(a = \frac{1}{5}\)[/tex].
2. Likewise, [tex]\(16z^2\)[/tex] is a perfect square as well. This can be expressed as [tex]\((4z)^2\)[/tex]. So here, [tex]\(b = 4z\)[/tex].
Now we can apply the difference of squares formula:
[tex]\[ \frac{1}{25} - 16z^2 = \left(\frac{1}{5}\right)^2 - (4z)^2 \][/tex]
This can be factored as:
[tex]\[ \left(\frac{1}{5} - 4z\right)\left(\frac{1}{5} + 4z\right) \][/tex]
This is the factored form of the expression. Therefore, the correct choice is:
A. [tex]\(\frac{1}{25}-16 z^2 = \left(\frac{1}{5} - 4z\right)\left(\frac{1}{5} + 4z\right)\)[/tex]
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