g) $\frac{25^{-1}}{5^{-2}} \div \frac{0,05^{-2}}{0,1^{-3}}$
Answer (2)
Rewrite the expression using fractions and decimals.
Simplify the exponents using exponent rules.
Deal with the negative exponents by taking reciprocals.
Perform the division to obtain the final answer: 2.5 .
Explanation
Understanding the Expression We are asked to simplify the expression 5 − 2 2 5 − 1 ÷ 0 , 1 − 3 0 , 0 5 − 2 . Let's break this down step by step.
Rewriting the Expression First, we can rewrite the expression using fractions and decimals. We know that 25 = 5 2 , 0.05 = 100 5 = 20 1 , and 0.1 = 10 1 . Substituting these values, we get: 5 − 2 ( 5 2 ) − 1 ÷ ( 10 1 ) − 3 ( 20 1 ) − 2
Simplifying Exponents Now, let's simplify the exponents. Recall that ( a m ) n = a m × n and a − n = a n 1 . So, we have: 5 − 2 5 − 2 ÷ ( 10 1 ) − 3 ( 20 1 ) − 2 Since 5 − 2 5 − 2 = 1 , the expression simplifies to: 1 ÷ ( 10 1 ) − 3 ( 20 1 ) − 2
Dealing with Negative Exponents Next, we simplify the fraction with negative exponents. Recall that ( a 1 ) − n = a n . Thus, we have: ( 20 1 ) − 2 = 2 0 2 = 400 ( 10 1 ) − 3 = 1 0 3 = 1000 So, the expression becomes: 1 ÷ 1000 400
Performing the Division Now, we perform the division. Dividing by a fraction is the same as multiplying by its reciprocal. So, we have: 1 ÷ 1000 400 = 1 × 400 1000 = 400 1000 = 4 10 = 2 5 = 2.5
Final Answer Therefore, the simplified expression is 2.5 .
Examples
Understanding how to simplify expressions with exponents and fractions is useful in many areas, such as calculating growth rates, understanding compound interest, or converting units in physics. For example, if you are calculating the future value of an investment with compound interest, you might need to simplify an expression with exponents and fractions to find the final amount. Similarly, in physics, you might need to convert units from one system to another, which often involves simplifying expressions with exponents and fractions.
To simplify the expression 5 − 2 2 5 − 1 ÷ 0. 1 − 3 0.0 5 − 2 , we rewrite and simplify using exponent rules. After processing, we find that the final result is 2.5 . Understanding how to work with exponents and fractions is essential in math.
;
