Date
itive exponents.
2) $\frac{3}{3^3}$
Answer (1)
Evaluate the denominator: 3 3 = 27 , so the expression becomes 27 3 .
Simplify the fraction: 27 3 = 9 1 .
Alternatively, use exponent rules: 3 3 3 1 = 3 1 − 3 = 3 − 2 = 3 2 1 = 9 1 .
The simplified expression with positive exponents is 9 1 .
Explanation
Understanding the Problem We are given the expression 3 3 3 and asked to simplify it, ensuring the final answer has positive exponents.
Evaluating the Denominator First, let's evaluate the denominator: 3 3 = 3 × 3 × 3 = 27 . So the expression becomes 27 3 .
Simplifying the Fraction Now, we can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3. Thus, 27 3 = 27 ÷ 3 3 ÷ 3 = 9 1 .
Using Exponent Rules Alternatively, we can use the properties of exponents. We can rewrite the expression as 3 3 3 1 . Using the rule a n a m = a m − n , we have 3 1 − 3 = 3 − 2 . To express this with a positive exponent, we use the rule a − n = a n 1 , so 3 − 2 = 3 2 1 = 9 1 .
Final Answer Therefore, the simplified expression with positive exponents is 9 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 $1000 at an annual interest rate of 5%, compounded annually, the amount you have after 3 years can be calculated as 1000 ( 1 + 0.05 ) 3 . Simplifying expressions with exponents helps in determining the final amount. Similarly, in scientific notation, simplifying exponents is crucial for handling very large or very small numbers, making calculations more manageable.
