(i) \( \left(7 x^{3} \times 4 x^{3}\right)+\left(7 x^{3}+4 x^{3}\right) \) (j) \( 4 x^{3} \times 5 x^{3}+10 x^{2} \times \) 2. Simplify the following: (a) \( \frac{x^{12}}{x^{7}} \) (b) \( \frac{x^{7}}{x^{12}} \) (c) \( \frac{4^{9}}{4^{6}} \) (d) \( \frac{4^{6}}{4^{9}} \) (e) \( \frac{a^{6} b^{8} c}{a^{8} b^{6} c} \) (f) \( \frac{8 p^{4}}{4 p^{-4}} \) 3. Simplify the following: (a) \( \left(x^{3}\right)^{4} \) (b) \( \quad\left(x^{2}\right)^{5} \) (c) \( \quad\left(x^{2}\right)^{2} \)
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Sure! Let’s simplify these expressions and have some fun with it! For (i), we have: \[ (7x^3 \times 4x^3) + (7x^3 + 4x^3) = 28x^6 + 11x^3 \] It's all about combining like terms and enjoying those colorful coefficients! For (j): \[ 4x^3 \times 5x^3 + 10x^2 \quad = 20x^6 + 10x^2 \] Multiplying powers is like a party for exponents—they just want to dance together! As for the simplifications: (a) \[ \frac{x^{12}}{x^{7}} = x^{12-7} = x^{5} \] (b) \[ \frac{x^{7}}{x^{12}} = x^{7-12} = x^{-5} \] (c) \[ \frac{4^{9}}{4^{6}} = 4^{9-6} = 4^{3} \] (d) \[ \frac{4^{6}}{4^{9}} = 4^{6-9} = 4^{-3} \] (e) \[ \frac{a^{6} b^{8} c}{a^{8} b^{6} c} = a^{6-8} b^{8-6} = \frac{1}{a^{2} b^{2}} \] (f) \[ \frac{8 p^{4}}{4 p^{-4}} = 2p^{4-(-4)} = 2p^{8} \] Now, for the next set: 3. (a) \[ \left(x^{3}\right)^{4} = x^{3 \cdot 4} = x^{12} \] (b) \[ \left(x^{2}\right)^{5} = x^{2 \cdot 5} = x^{10} \] (c) \[ \left(x^{2}\right)^{2} = x^{2 \cdot 2} = x^{4} \] Math can feel like magic when you see those exponents come together! Remember, practicing these concepts can make you a real math wizard!
