\frac{(-2)^8 \cdot 5^3}{5^4 \cdot 2^{10} \cdot 10}
Answer (1)
Rewrite the expression with positive exponents and prime factorization: 5 4 \t 2 10 \t ( 2 \t 5 ) 2 8 \t 5 3 .
Simplify the powers in the denominator: 5 5 \t 2 11 2 8 \t 5 3 .
Cancel out common factors: 2 3 \t 5 2 1 .
Calculate the final value: 200 1 .
Explanation
Understanding the Expression We are asked to simplify the expression 5 4 \t 2 10 \t 10 ( − 2 ) 8 \t 5 3 . Let's break this down step by step.
Rewriting the Expression First, we can rewrite ( − 2 ) 8 as 2 8 since a negative number raised to an even power is positive. Also, we can rewrite 10 as 2 \t 5 . So the expression becomes 5 4 \t 2 10 \t ( 2 \t 5 ) 2 8 \t 5 3 .
Simplifying Powers Now, we can simplify the denominator by combining the powers of 2 and 5. We have 5 4 \t 5 = 5 4 + 1 = 5 5 and 2 10 \t 2 = 2 10 + 1 = 2 11 . Thus, the expression is now 5 5 \t 2 11 2 8 \t 5 3 .
Canceling Common Factors Next, we can simplify the fraction by canceling out common factors. We have 2 11 2 8 = 2 11 − 8 1 = 2 3 1 and 5 5 5 3 = 5 5 − 3 1 = 5 2 1 . So the expression becomes 2 3 \t 5 2 1 .
Calculating Powers Now, we calculate 2 3 = 2 \t 2 \t 2 = 8 and 5 2 = 5 \t 5 = 25 . Thus, the expression is 8 \t 25 1 .
Final Calculation Finally, we calculate 8 \t 25 = 200 . Therefore, the simplified expression is 200 1 .
Examples
In manufacturing, you might need to calculate the efficiency of a process. If you have a certain amount of raw material and you want to find out how much usable product you get after several steps, you might end up with a fraction similar to the one we simplified. Simplifying such fractions helps in understanding the yield and optimizing the process to reduce waste and increase efficiency. For example, if you start with a certain amount of metal and, after melting, shaping, and polishing, you end up with a fraction representing the usable metal, simplifying that fraction gives you a clear picture of your process's efficiency.
