$\frac{5^x-5^{x-1}}{4 \times 5^{x-1}}$
Answer (1)
Rewrite 5 x as 5 \tims 5 x − 1 .
Factor out 5 x − 1 from the numerator: 4 × 5 x − 1 5 x − 1 ( 5 − 1 ) .
Simplify the expression: 4 × 5 x − 1 5 x − 1 ( 4 ) .
Cancel out common factors: 4 × 5 x − 1 4 × 5 x − 1 = 1 . The simplified expression is 1 .
Explanation
Understanding the Problem We are asked to simplify the expression 4 × 5 x − 1 5 x − 5 x − 1 . To do this, we will factor out 5 x − 1 from the numerator and then simplify the resulting expression.
Rewriting the Expression First, let's rewrite 5 x as 5 × 5 x − 1 . This allows us to factor out 5 x − 1 from both terms in the numerator:
4 × 5 x − 1 5 x − 5 x − 1 = 4 × 5 x − 1 5 × 5 x − 1 − 5 x − 1
Factoring the Numerator Now, we factor out 5 x − 1 from the numerator:
4 × 5 x − 1 5 × 5 x − 1 − 5 x − 1 = 4 × 5 x − 1 5 x − 1 ( 5 − 1 )
Simplifying the Parentheses Simplify the expression inside the parentheses:
4 × 5 x − 1 5 x − 1 ( 5 − 1 ) = 4 × 5 x − 1 5 x − 1 ( 4 )
Canceling Common Factors Now, we can cancel out the common factors of 4 and 5 x − 1 from the numerator and the denominator:
4 × 5 x − 1 5 x − 1 ( 4 ) = 4 × 5 x − 1 4 × 5 x − 1 = 1
Final Answer Therefore, the simplified expression is 1.
Examples
Simplifying exponential expressions is useful in various fields, such as calculating growth rates in finance or modeling radioactive decay in physics. For example, if you want to determine the remaining amount of a radioactive substance after a certain time, you might encounter an expression similar to the one we simplified. By simplifying such expressions, you can easily calculate the desired quantity without complex calculations. This skill is also valuable in computer science when analyzing algorithms with exponential time complexity.
