$\frac{625 \times 5^{-50}}{25^2}$
Answer (2)
The expression 2 5 2 625 × 5 − 50 simplifies to 5 − 50 by expressing 625 and 25 as powers of 5, simplifying the numerator and denominator, and then subtracting the exponents. Therefore, the final answer is 5 − 50 .
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Express 625 and 25 as powers of 5: 625 = 5 4 and 25 = 5 2 .
Substitute into the expression: ( 5 2 ) 2 5 4 × 5 − 50 .
Simplify the denominator: ( 5 2 ) 2 = 5 4 .
Simplify the expression: 5 4 5 4 × 5 − 50 = 5 − 50 .
5 − 50
Explanation
Problem Analysis We are given the expression 2 5 2 625 × 5 − 50 and our goal is to simplify it.
Expressing as Powers of 5 First, we express 625 and 25 as powers of 5. We know that 625 = 5 4 and 25 = 5 2 . Substituting these into the expression, we get ( 5 2 ) 2 5 4 × 5 − 50 .
Simplifying the Denominator Next, we simplify the denominator. ( 5 2 ) 2 = 5 2 × 2 = 5 4 . So the expression becomes 5 4 5 4 × 5 − 50 .
Simplifying the Numerator Now, we simplify the numerator. 5 4 × 5 − 50 = 5 4 + ( − 50 ) = 5 − 46 . The expression is now 5 4 5 − 46 .
Final Simplification Finally, we simplify the entire expression by subtracting the exponents: 5 4 5 − 46 = 5 − 46 − 4 = 5 − 50 .
Final Answer Therefore, the simplified expression is 5 − 50 .
Examples
Understanding exponential simplification is crucial in various fields, such as computer science when dealing with memory or storage calculations. For instance, if you are analyzing the efficiency of a data storage system where the capacity is expressed in powers of 2 (since computers use binary), simplifying expressions like the one above can help determine how much data can be stored or processed. This skill is also valuable in physics when dealing with very large or very small quantities, such as the size of atoms or the distance between stars, which are often expressed using scientific notation involving exponents.
