Simplify each expression. $(35m^4n)\left(-\frac{2}{7}mn^2\right)$

Verified step by step guidance

1. Identify the expression to simplify: \( (35m^{4}n) \times \left(-\frac{2}{7}mn^{2}\right) \).
2. Multiply the coefficients (numerical parts) together: \( 35 \times \left(-\frac{2}{7}\right) \).
3. Apply the product rule for exponents to the variables with the same base: For \(m\), multiply \(m^{4} \times m^{1} = m^{4+1} = m^{5}\); for \(n\), multiply \(n^{1} \times n^{2} = n^{1+2} = n^{3}\).
4. Combine the results from the coefficients and variables to write the simplified expression as: \( \text{(coefficient result)} \times m^{5} n^{3} \).
5. Leave the expression in simplified form without calculating the final numerical value, as per instructions.

Key Concepts

Multiplication of Monomials

When multiplying monomials, multiply their coefficients (numerical parts) and then multiply variables by adding their exponents if the bases are the same. For example, (a^m)(a^n) = a^(m+n). This rule helps simplify expressions involving variables raised to powers.

Properties of Exponents

The properties of exponents include rules like product of powers, power of a power, and power of a product. Specifically, when multiplying like bases, add the exponents. This is essential for simplifying expressions with variables raised to powers.

Multiplication of Fractions

To multiply fractions, multiply the numerators together and the denominators together. Simplify the resulting fraction by reducing common factors. This is important when coefficients are fractions, as in the given expression.