Question
Activity 4: Fully simplify the following ex \( \begin{array}{ll}4.1 & \left(3 a^{2} b\right)^{3} \\ 4.2 & \left(\frac{2}{3} x^{3} y^{2}\right)^{4} \\ 4.3 & 2\left(-3 m n^{2}\right)^{2}\end{array} \)
Ask by Macdonald Lyons. in South Africa
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
4.1: 27a⁶b³
4.2: (16/81)x¹²y⁸
4.3: 18m²n⁴
Solution
Let's simplify each expression step by step.
4.1: Simplify (3a²b)³
• Rewrite by raising each part to the power of 3:
(3a²b)³ = 3³ · (a²)³ · b³
• Compute each term:
3³ = 27
(a²)³ = a^(2·3) = a⁶
b³ remains b³
• Thus, the simplified form is:
27a⁶b³
4.2: Simplify ((2/3)x³y²)⁴
• Distribute the exponent to each factor:
((2/3)x³y²)⁴ = (2/3)⁴ · (x³)⁴ · (y²)⁴
• Compute each term:
(2/3)⁴ = 2⁴/3⁴ = 16/81
(x³)⁴ = x^(3·4) = x¹²
(y²)⁴ = y^(2·4) = y⁸
• The final simplified expression is:
(16/81)x¹²y⁸
4.3: Simplify 2(-3mn²)²
• First, simplify (-3mn²)² by applying the exponent:
(-3mn²)² = (-3)² · (m)² · (n²)²
• Compute each part:
(-3)² = 9
(m)² = m²
(n²)² = n^(2·2) = n⁴
• Thus, (-3mn²)² = 9m²n⁴
• Now multiply by the 2 outside:
2 · 9m²n⁴ = 18m²n⁴
Final Answers:
• 4.1: 27a⁶b³
• 4.2: (16/81)x¹²y⁸
• 4.3: 18m²n⁴
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution The Deep Dive
When simplifying \( (3 a^{2} b)^{3} \), you want to raise each part to the power of three. So, you get \( 3^{3} (a^{2})^{3} b^{3} = 27 a^{6} b^{3} \). For \( \left(\frac{2}{3} x^{3} y^{2}\right)^{4} \), apply the power of four to everything inside the parentheses. This results in \( \left(\frac{2^{4}}{3^{4}}\right)x^{12} y^{8} = \frac{16}{81} x^{12} y^{8} \). Finally, to simplify \( 2(-3 m n^{2})^{2} \), first calculate \( (-3 m n^{2})^{2} = 9 m^{2} n^{4} \), then multiply by 2. So, you get \( 2 \cdot 9 m^{2} n^{4} = 18 m^{2} n^{4} \).
