$\frac{27^{-6}}{125^{-3}} \div \frac{9^{-2}}{25^{-4}}$
Answer (1)
Rewrite the numbers as powers of 3 and 5.
Substitute these into the expression and simplify the exponents.
Rewrite the division as multiplication by the reciprocal and combine the powers.
Calculate the final simplified expression: 4782969 5
Explanation
Understanding the Problem We are given the expression 12 5 − 3 2 7 − 6 ÷ 2 5 − 4 9 − 2 and we need to simplify it. The numbers 27, 125, 9, and 25 can be written as powers of prime numbers 3 and 5.
Rewriting as Powers of Primes First, rewrite the numbers as powers of 3 and 5: 27 = 3 3 125 = 5 3 9 = 3 2 25 = 5 2
Substituting and Simplifying Exponents Substitute these into the expression: ( 5 3 ) − 3 ( 3 3 ) − 6 ÷ ( 5 2 ) − 4 ( 3 2 ) − 2 Simplify the exponents: 5 − 9 3 − 18 ÷ 5 − 8 3 − 4
Division as Multiplication Rewrite the division as multiplication by the reciprocal: 5 − 9 3 − 18 × 3 − 4 5 − 8 Combine the powers of 3 and 5: 3 − 18 − ( − 4 ) × 5 − 8 − ( − 9 ) Simplify the exponents: 3 − 14 × 5 1
Final Simplification Rewrite the expression: 3 14 5 Calculate 3 14 . The result of the operation is 4782969.
Final Answer So the final simplified expression is: 4782969 5
Examples
Understanding how to simplify expressions with exponents is useful in many areas, such as calculating compound interest or dealing with scientific notation in physics or chemistry. For example, if you are calculating the future value of an investment with continuously compounded interest, you might encounter expressions with exponents. Simplifying these expressions allows you to easily determine the final value of your investment. Similarly, in scientific calculations, simplifying expressions with exponents helps in managing very large or very small numbers efficiently.
