Simplify the expression: [tex]0.2 a^2 b^3 \cdot\left(-5 a^3 b\right)^2[/tex]
Answer (1)
First, square the term inside the parenthesis: ( − 5 a 3 b ) 2 = 25 a 6 b 2 .
Then, multiply the result with the remaining term: 0.2 a 2 b 3 ⋅ 25 a 6 b 2 .
Simplify the numerical coefficients: 0.2 ⋅ 25 = 5 .
Simplify the variable terms using the exponent rule: a 2 ⋅ a 6 = a 8 and b 3 ⋅ b 2 = b 5 , so the final answer is 5 a 8 b 5 .
Explanation
Understanding the Problem We are asked to simplify the expression 0.2 a 2 b 3 ⋅ ( − 5 a 3 b ) 2 . This involves applying exponent rules and combining like terms.
Simplifying the Parenthetical Term First, we need to simplify the term inside the parenthesis that is raised to the power of 2. We have ( − 5 a 3 b ) 2 . Applying the power rule, we get: ( − 5 a 3 b ) 2 = ( − 5 ) 2 ( a 3 ) 2 ( b ) 2 = 25 a 3 × 2 b 2 = 25 a 6 b 2
Substituting and Rearranging Now, we substitute this back into the original expression: 0.2 a 2 b 3 ⋅ ( 25 a 6 b 2 ) Next, we multiply the numerical coefficients and combine the like terms: 0.2 × 25 × a 2 × a 6 × b 3 × b 2
Combining Like Terms We know that 0.2 × 25 = 5 . Also, when multiplying terms with the same base, we add the exponents. So, a 2 × a 6 = a 2 + 6 = a 8 and b 3 × b 2 = b 3 + 2 = b 5 . Therefore, the expression simplifies to: 5 a 8 b 5
Final Answer Thus, the simplified expression is 5 a 8 b 5 .
Examples
In physics, when calculating the kinetic energy of a rotating object, you might encounter expressions similar to the one simplified here. For instance, if the moment of inertia involves terms with variables raised to certain powers, simplifying such expressions helps in determining the object's energy more efficiently. Suppose the moment of inertia I is proportional to 0.2 m 2 r 3 and the angular velocity ω is proportional to ( − 5 m 3 r ) 2 , where m is mass and r is radius. The kinetic energy K = 2 1 I ω 2 would then involve simplifying the expression we just worked with, making the calculation straightforward.
