\frac{(-2)^8 \cdot 5^3}{5^4 \cdot 2^{10} \cdot 10}
Answer (1)
The goal is to simplify the expression 5 4 ⋅ 2 10 ⋅ 10 ( − 2 ) 8 ⋅ 5 3 .
Rewrite the expression as 5 4 ⋅ 2 10 ⋅ ( 2 ⋅ 5 ) 2 8 ⋅ 5 3 .
Combine the powers: 2 11 ⋅ 5 5 2 8 ⋅ 5 3 .
Cancel out common factors: 2 3 ⋅ 5 2 1 .
Calculate the powers and multiply: 200 1 .
The final answer is 200 1 .
Explanation
Rewrite the expression We are asked to simplify the expression 5 4 ⋅ 2 10 ⋅ 10 ( − 2 ) 8 ⋅ 5 3 . Let's break this down step by step. First, we observe that 10 = 2 ⋅ 5 . Also, since the exponent 8 is even, ( − 2 ) 8 = 2 8 . So we can rewrite the expression as 5 4 ⋅ 2 10 ⋅ ( 2 ⋅ 5 ) 2 8 ⋅ 5 3 .
Combine the powers Now, let's combine the powers of 2 and 5 in the denominator. We have 2 10 ⋅ 2 = 2 11 and 5 4 ⋅ 5 = 5 5 . So the expression becomes 2 11 ⋅ 5 5 2 8 ⋅ 5 3 .
Cancel common factors Next, we can cancel out the common factors of 2 and 5. We have 2 11 2 8 = 2 8 − 11 = 2 − 3 = 2 3 1 and 5 5 5 3 = 5 3 − 5 = 5 − 2 = 5 2 1 . So the expression simplifies to 2 3 ⋅ 5 2 1 .
Calculate the powers Now, let's calculate 2 3 and 5 2 . We have 2 3 = 2 ⋅ 2 ⋅ 2 = 8 and 5 2 = 5 ⋅ 5 = 25 . So the expression becomes 8 ⋅ 25 1 .
Multiply the numbers Finally, let's multiply 8 and 25. We have 8 ⋅ 25 = 200 . So the simplified expression is 200 1 .
Examples
Understanding how to simplify expressions with exponents is useful in many areas, such as calculating compound interest or dealing with scientific notation. For example, if you invest money with a certain annual interest rate, you might need to simplify an expression involving exponents to find the total amount you'll have after a certain number of years. Simplifying expressions helps to make these calculations easier and more manageable, allowing you to understand the growth of your investment more clearly. Also, in physics, simplifying expressions with exponents is crucial when dealing with very large or very small numbers, such as the size of atoms or the distance to stars.
