[tex]\sqrt{25 a^2 b^2} \times \sqrt[3]{27 a^3}[/tex]
Answer (2)
Simplify the square root term: 25 a 2 b 2 = 5∣ a ∣∣ b ∣ .
Simplify the cube root term: 3 27 a 3 = 3 a .
Multiply the simplified terms: 5∣ a ∣∣ b ∣ × 3 a = 15 a ∣ a ∣∣ b ∣ .
The simplified expression is 15 a ∣ a ∣∣ b ∣ .
Explanation
Understanding the Problem We are asked to simplify the expression 25 a 2 b 2 × 3 27 a 3 . We will simplify each radical term separately and then multiply the results.
Simplifying the Square Root First, let's simplify the square root term 25 a 2 b 2 . We can rewrite this as 25 × a 2 × b 2 . Since 25 = 5 , a 2 = ∣ a ∣ , and b 2 = ∣ b ∣ , we have 25 a 2 b 2 = 5∣ a ∣∣ b ∣ .
Simplifying the Cube Root Next, let's simplify the cube root term 3 27 a 3 . We can rewrite this as 3 27 × 3 a 3 . Since 3 27 = 3 and 3 a 3 = a , we have 3 27 a 3 = 3 a .
Multiplying the Simplified Terms Now, we multiply the simplified terms: 5∣ a ∣∣ b ∣ × 3 a = 15 a ∣ a ∣∣ b ∣ .
Final Answer Therefore, the simplified expression is 15 a ∣ a ∣∣ b ∣ .
Examples
Simplifying radical expressions is useful in various fields such as physics and engineering. For example, when calculating the energy of a particle in quantum mechanics, you might encounter expressions involving square roots and cube roots of variables. Simplifying these expressions allows for easier manipulation and calculation of the energy levels.
To simplify the expression 25 a 2 b 2 × 3 27 a 3 , we find that 25 a 2 b 2 = 5∣ a ∣∣ b ∣ and 3 27 a 3 = 3 a . Multiplying these results gives 15 a ∣ a ∣∣ b ∣ as the final simplified expression.
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