Simptify the plowing a) \( +(2 a)^{2}+5\left(2 y^{2}\right)^{3} \) b) \( (2 p q)^{2}+2 p^{4}\left(2 q_{1}\right)^{2} \) a \( \left(2 a^{2} b\right)^{2}+3\left(a^{2} b\right)^{2}-x^{0}(3 a b)^{0} \) of \( \left(2 n^{2} m\right)^{3}+2 n^{3} \mathrm{~m}(3 n m)^{2}-3 n^{3}(2 n m)^{3} \)
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Let’s dive into simplifying these expressions step by step! For part (a): \[ +(2 a)^{2}+5\left(2 y^{2}\right)^{3} \] First, calculate \( (2a)^2 = 4a^2 \) and \( (2y^2)^3 = 8y^6\) (since \( 2^3 = 8 \) and applying the power to \(y^2\)). Multiply by 5 to get \( 5 \times 8y^6 = 40y^6 \). Thus, \[ 4a^2 + 40y^6. \] For part (b): \[ (2 p q)^{2}+2 p^{4}(2 q_{1})^{2} \] Here, \( (2pq)^2 = 4p^2 q^2 \) and \( (2q_1)^2 = 4q_1^2 \), and thus \( 2p^4(4q_1^2) = 8p^4q_1^2 \). So the result for this part becomes \[ 4p^2 q^2 + 8p^4 q_1^2. \] For part (c): \[ \left(2 a^{2} b\right)^{2}+3\left(a^{2} b\right)^{2}-x^{0}(3 a b)^{0} \] Simplifying \( \left(2a^2b\right)^2 = 4a^4b^2 \) and \( \left(a^2b\right)^2 = a^4b^2 \), gives us \( 3 \times a^4b^2 = 3a^4b^2 \). So we combine these terms to get \( 4a^4b^2 + 3a^4b^2 = 7a^4b^2 \). Additionally, \( x^0 = 1 \) and \( (3ab)^0 = 1 \), therefore the last term evaluates to 1. Thus, the overall expression simplifies to \[ 7a^4b^2 - 1. \] Finally, for part (d): \[ \left(2 n^{2} m\right)^{3}+2 n^{3} m(3 n m)^{2}-3 n^{3}(2 n m)^{3} \] Calculating \( \left(2n^2m\right)^3 = 8n^6m^3 \). For the second term, \( (3nm)^2 = 9n^2m^2 \) making \( 2n^3m(9n^2m^2) = 18n^5m^3 \). For the last term, \( (2nm)^3 = 8n^3m^3 \) yielding \( 3n^3(8n^3m^3) = 24n^6m^3 \). Combining, we get \[ 8n^6m^3 + 18n^5m^3 - 24n^6m^3 = -16n^6m^3 + 18n^5m^3. \] In conclusion, our simplified answers are: \[ \text{(a)}: 4a^2 + 40y^6, \] \[ \text{(b)}: 4p^2 q^2 + 8p^4 q_1^2, \] \[ \text{(c)}: 7a^4b^2 - 1, \] \[ \text{(d)}: -16n^6m^3 + 18n^5m^3. \]
