(5x^3y/20xy^5)^4
According to my teacher the answer is x^8/256y^16, but I don't understand how to get there.
To simplify the expression (5x^3y/20xy^5)^4, we can break it down step by step.
Step 1: Simplify the numerator and denominator individually.
- Numerator: 5x^3y
- Denominator: 20xy^5
Step 2: Simplify the numerator and denominator separately using exponent rules.
- Numerator: (5x^3y)^4 = 5^4 * (x^3)^4 * y^4 = 625x^12y^4
- Denominator: (20xy^5)^4 = 20^4 * (x)^4 * (y^5)^4 = 160,000x^4y^20
Step 3: Divide the simplified numerator by the simplified denominator.
- Final answer: (625x^12y^4)/(160,000x^4y^20)
To further simplify the expression, we can divide the common factors between the numerator and denominator.
Step 4: Divide the coefficients (numbers) and simplify the exponents.
- Coefficients: 625/160,000 can be simplified by dividing both numbers by 25.
625/25 = 25 and 160,000/25 = 6,400.
- Variables: x^12/x^4 = x^(12-4) = x^8
y^4/y^20 = y^(4-20) = y^(-16), where any non-zero number raised to a negative exponent is equal to 1 divided by that number raised to the positive exponent.
Step 5: Write the final answer with the simplified expression.
- Final Answer: (625/160,000) * x^8 * y^(-16) = x^8/256 * (1/y^16) = x^8/256y^16
Therefore, the final answer is x^8/256y^16, which matches the answer given by your teacher.