Simplify each exponential expression in Exercises 23–64. $\left(3x^4\right)\left(2x^7\right)$

Verified step by step guidance

1. Identify the properties of exponents that apply to the expression \((3x^4)(2x^7)\). Specifically, recall that when multiplying terms with the same base, you add the exponents.
2. Rewrite the expression by separating the coefficients and the variable parts: \((3)(2) \times (x^4)(x^7)\).
3. Multiply the coefficients: \$3 \times 2$.
4. Apply the product of powers property to the variable part: \(x^{4+7}\).
5. Combine the results to write the simplified expression as the product of the multiplied coefficients and the variable with the new exponent.

Key Concepts

Product of Powers Property

This property states that when multiplying two expressions with the same base, you add their exponents. For example, x^a * x^b = x^(a+b). This rule helps simplify expressions like x^4 * x^7 by combining the exponents.

Multiplying Coefficients

When multiplying terms, multiply the numerical coefficients separately from the variables. For instance, in (3x^4)(2x^7), multiply 3 and 2 to get 6 before applying exponent rules to the variable parts.

Simplifying Exponential Expressions

Simplifying exponential expressions involves applying exponent rules and combining like terms to write the expression in its simplest form. This process makes expressions easier to work with and understand.