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The expression \( \frac{m^{2}}{9} - 64 \) can be factored as \( \left(\frac{m}{3} - 8\right)\left(\frac{m}{3} + 8\right) \).
Solution
1. Write the expression as a difference of squares:
\[
\frac{m^2}{9} - 64 = \left(\frac{m}{3}\right)^2 - 8^2
\]
2. Factor using the difference of squares formula:
\[
a^2 - b^2 = (a - b)(a + b)
\]
where \(a = \frac{m}{3}\) and \(b = 8\). Thus:
\[
\left(\frac{m}{3} - 8\right)\left(\frac{m}{3} + 8\right)
\]
The factorized form is:
\[
\left(\frac{m}{3} - 8\right)\left(\frac{m}{3} + 8\right)
\]
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Simplify this solution The Deep Dive
To factor the expression \( \frac{m^{2}}{9} - 64 \), we can recognize that it's a difference of squares. We can rewrite the expression as \( \left(\frac{m}{3}\right)^{2} - 8^{2} \), which can be factored using the formula \( a^{2} - b^{2} = (a - b)(a + b) \). Thus, it becomes \( \left(\frac{m}{3} - 8\right)\left(\frac{m}{3} + 8\right) \). Next, let’s check for common mistakes when working with differences of squares. A frequent error is forgetting to recognize it as a difference of squares altogether! Always look for terms that are perfect squares and remember to apply the correct signs. It’s easy to overlook the layout, but with practice, you’ll spot them like a pro!
