Question
1. \( (x+3)(x-3)= \) 2. \( (3 x+5 y)(3 x-5 y)= \) 3. \( \left(6 x^{5}+1\right)\left(6 x^{5}-1\right)= \) 4. \( \left(9 a b^{4}-c^{7}\right)\left(9 a b^{4}+c^{7}\right)= \) 5. \( \left(\frac{3}{5} m+\frac{1}{2}\right)\left(\frac{3}{5} m-\frac{1}{2}\right)= \) 6. \( \left(\frac{7}{6} x^{3}+\frac{3}{2}\right)\left(\frac{7}{6} x^{3}-\frac{3}{2}\right)= \) 7. \( \left(\frac{1}{3} x y+z^{6}\right)\left(\frac{1}{3} x y-z^{6}\right)= \) 8. \( \left(3 x^{2}+\frac{1}{10}\right)\left(-3 x^{2}+\frac{1}{10}\right)= \)
Ask by Carter Patel. in Mexico
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
1. \( (x + 3)(x - 3) = x^{2} - 9 \)
2. \( (3x + 5y)(3x - 5y) = 9x^{2} - 25y^{2} \)
3. \( (6x^{5} + 1)(6x^{5} - 1) = 36x^{10} - 1 \)
4. \( (9ab^{4} - c^{7})(9ab^{4} + c^{7}) = 81a^{2}b^{8} - c^{14} \)
5. \( \left(\frac{3}{5}m + \frac{1}{2}\right)\left(\frac{3}{5}m - \frac{1}{2}\right) = \frac{9}{25}m^{2} - \frac{1}{4} \)
6. \( \left(\frac{7}{6}x^{3} + \frac{3}{2}\right)\left(\frac{7}{6}x^{3} - \frac{3}{2}\right) = \frac{49}{36}x^{6} - \frac{9}{4} \)
7. \( \left(\frac{1}{3}xy + z^{6}\right)\left(\frac{1}{3}xy - z^{6}\right) = \frac{1}{9}x^{2}y^{2} - z^{12} \)
8. \( \left(3x^{2} + \frac{1}{10}\right)\left(-3x^{2} + \frac{1}{10}\right) = -9x^{4} + \frac{1}{100} \)
Solution
Claro, a continuación se expanden las expresiones dadas:
1. \( (x + 3)(x - 3) = x^{2} - 9 \)
2. \( (3x + 5y)(3x - 5y) = 9x^{2} - 25y^{2} \)
3. \( (6x^{5} + 1)(6x^{5} - 1) = 36x^{10} - 1 \)
4. \( (9ab^{4} - c^{7})(9ab^{4} + c^{7}) = 81a^{2}b^{8} - c^{14} \)
5. \( \left(\frac{3}{5}m + \frac{1}{2}\right)\left(\frac{3}{5}m - \frac{1}{2}\right) = \frac{9}{25}m^{2} - \frac{1}{4} \)
6. \( \left(\frac{7}{6}x^{3} + \frac{3}{2}\right)\left(\frac{7}{6}x^{3} - \frac{3}{2}\right) = \frac{49}{36}x^{6} - \frac{9}{4} \)
7. \( \left(\frac{1}{3}xy + z^{6}\right)\left(\frac{1}{3}xy - z^{6}\right) = \frac{1}{9}x^{2}y^{2} - z^{12} \)
8. \( \left(3x^{2} + \frac{1}{10}\right)\left(-3x^{2} + \frac{1}{10}\right) = -9x^{4} + \frac{1}{100} \)
Estas expansiones se han realizado utilizando la identidad de la diferencia de cuadrados, que establece que \((a + b)(a - b) = a^{2} - b^{2}\).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
The expressions you've got there represent a classic math concept: the difference of squares. This nifty mathematical formulation helps simplify calculations by turning a product of a sum and a difference into a difference of squares. The result is often a quadratic or a higher-degree polynomial that can be quite easier to work with than the original expression. For example, \( (a + b)(a - b) = a^2 - b^2 \). This means when you see something in the form of \( (x + n)(x - n) \) or \( (3x + m)(3x - m) \), you can straight-up compute \( a^2 - b^2 \) where \( a \) and \( b \) are your expressions. It’s like math’s simplified cheat code for squaring things away quickly!
