Final answer:
To simplify the given expression, we can rewrite the terms in the numerator and denominator using the rules of exponents, then cancel out the common factors to simplify further. Finally, we can apply the rules of exponentiation to simplify the expression to its final form, which is 625x^9.
Step-by-step explanation:
To simplify the expression (125 × x^(-3)) / (5^(-3) × 25 × x^(-6)), we can start by simplifying the terms in the numerator and denominator. In the numerator, 125 × x^(-3) can be written as (5^3) × x^(-3). In the denominator, 5^(-3) × 25 × x^(-6) can be written as (5^(-3)) × (5^2) × x^(-6). Now, we can cancel out the common factors and simplify the expression.
- First, let's simplify the numerator: (5^3) × x^(-3) = 125x^(-3)
- Next, let's simplify the denominator: (5^(-3)) × (5^2) × x^(-6) = (1/125) × 25 × x^(-6) = (25/125) × x^(-6) = (1/5) × x^(-6) = (x^(-6))/5
- Now, let's substitute these simplified terms back into the original expression: (125x^(-3))/((x^(-6))/5)
- When we divide by a fraction, we can multiply by its reciprocal. So, we can rewrite the expression as: (125x^(-3)) × (5/(x^(-6)))
- To simplify further, we can multiply the constants together: 125 × 5 = 625
- For the variables, we can apply the rule of exponentiation by subtracting the exponents: x^(-3) × x^6 = x^(6-3) = x^3
- So, the simplified expression is: 625x^3/(x^(-6))
- Finally, we can simplify further by applying the rule of exponentiation: x^(-6) = 1/x^6
- Substituting this back into the expression, we get the final simplified form: 625x^3 / (1/x^6) = 625x^3 * x^6 = 625x^(3+6) = 625x^9
Therefore, the simplified form of the expression is 625x^9.