Simplify the expression: [tex]$\left(0.2 m^2 n\right)^3 \cdot 1000 m^4 n^7$[/tex]
Answer (2)
Rewrite 0.2 as 5 1 .
Apply the power of a product rule to get ( 5 1 ) 3 ( m 2 ) 3 ( n ) 3 ⋅ 1000 m 4 n 7 .
Apply the power of a power rule to get 125 1 m 6 n 3 ⋅ 1000 m 4 n 7 .
Simplify the expression to get 8 m 10 n 10 .
Explanation
Rewrite 0.2 as a fraction We are asked to simplify the expression ( 0.2 m 2 n ) 3 ⋅ 1000 m 4 n 7 . Let's start by rewriting 0.2 as a fraction, which is 5 1 . So the expression becomes ( 5 1 m 2 n ) 3 ⋅ 1000 m 4 n 7 .
Apply the power of a product rule Next, we apply the power of a product rule, which states that ( ab ) n = a n b n . Applying this to the first term, we get ( 5 1 ) 3 ( m 2 ) 3 ( n ) 3 ⋅ 1000 m 4 n 7 .
Apply the power of a power rule Now, we apply the power of a power rule, which states that ( a m ) n = a mn . So, ( m 2 ) 3 = m 2 ⋅ 3 = m 6 . Thus, the expression becomes ( 5 1 ) 3 m 6 n 3 ⋅ 1000 m 4 n 7 .
Calculate (1/5)^3 We know that ( 5 1 ) 3 = 5 3 1 = 125 1 . So the expression is 125 1 m 6 n 3 ⋅ 1000 m 4 n 7 .
Multiply the terms Now, we multiply the terms together. 125 1 ⋅ 1000 = 125 1000 = 8 . Also, m 6 ⋅ m 4 = m 6 + 4 = m 10 , and n 3 ⋅ n 7 = n 3 + 7 = n 10 .
Final Answer Therefore, the simplified expression is 8 m 10 n 10 .
Examples
Understanding how to simplify expressions with exponents is crucial in many fields, such as physics and engineering. For example, when calculating the volume of a cube with sides that are growing over time, you might encounter expressions similar to the one we just simplified. Simplifying these expressions allows engineers to easily predict how the volume changes with respect to time or other variables, which is essential for designing structures or systems that can adapt to changing conditions. This skill also helps in financial modeling, where understanding exponential growth or decay is vital for predicting investment returns or loan payments.
The expression ( 0.2 m 2 n ) 3 ⋅ 1000 m 4 n 7 simplifies to 8 m 10 n 10 by rewriting and applying the power rules for exponents and combining like terms.
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