simplify -5y^4(y^5)^2 /15y^7(y^2)^3
To simplify the expression -5y^4(y^5)^2 / 15y^7(y^2)^3, we can follow the order of operations (also known as PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and use the properties of exponents.
Step 1: Simplify within the parentheses.
Inside the first parentheses, we have (y^5)^2. By raising a power to another exponent, we multiply the exponents, so (y^5)^2 becomes y^(5*2) which simplifies to y^10.
Inside the second parentheses, we have (y^2)^3. Similarly, (y^2)^3 becomes y^(2*3) which simplifies to y^6.
Our expression now becomes -5y^4 * y^10 / 15y^7 * y^6.
Step 2: Simplify the exponent when dividing with the same base.
When dividing with the same base, we subtract the exponents, so y^10 / y^7 becomes y^(10-7) which simplifies to y^3.
Similarly, y^6 / y^4 becomes y^(6-4) which simplifies to y^2.
Our expression now becomes -5y^4 * y^3 / 15y^2.
Step 3: Simplify the coefficient.
We know that -5 / 15 can be simplified to -1 / 3.
Our expression now becomes (-1/3) * y^4 * y^3 / y^2.
Step 4: Simplify the exponents using the product rule.
When multiplying with the same base, we add the exponents, so y^4 * y^3 becomes y^(4+3) which simplifies to y^7.
Our expression now becomes (-1/3) * y^7 / y^2.
Step 5: Simplify the exponent when dividing with the same base.
Dividing with the same base means we subtract the exponents, so y^7 / y^2 becomes y^(7-2) which simplifies to y^5.
Our simplified expression is now (-1/3) * y^5.