Product and Power Rules

The Product Rule states that when multiplying two expressions, you can multiply their coefficients and add their exponents if the bases are the same.

To simplify (9h)(-4h), multiply the coefficients (9 * -4 = -36) and add the exponents of h (h^1 * h^1 = h^(1+1) = h^2), resulting in -36h^2.

The Power Rule states that when raising a power to another power, you multiply the exponents.

To simplify (4qm^2p^2)^4, raise the coefficient to the fourth power (4^4 = 256) and multiply the exponents of each variable by 4, resulting in 256q^4m^8p^8.

To simplify (4xy^3)^3, raise the coefficient to the third power (4^3 = 64) and multiply the exponents of each variable by 3, resulting in 64x^3y^9.

To simplify (2a^2b^4z)(6a^3b^2z^5), multiply the coefficients (2 * 6 = 12) and add the exponents of each variable: a^(2+3), b^(4+2), z^(1+5), resulting in 12a^5b^6z^6.

To simplify (3y^4)^2, square the coefficient (3^2 = 9) and multiply the exponent of y by 2, resulting in 9y^8.

If f(x) = g(x) * h(x), then the derivative f'(x) = g'(x) * h(x) + g(x) * h'(x).

Raising a power to a power means multiplying the exponents. For example, (a^m)^n = a^(m*n).

When multiplying a positive coefficient by a negative coefficient, the result is negative. For example, 9 * -4 = -36.