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In algebra, a coefficient is a numerical value that is multiplied by a variable in an expression. For example, in the term \( 0.5a^3 \), the number 0.5 is the coefficient. To simplify an expression involving multiple terms, you need to first multiply the coefficients together.

Let's look at our example \( 0.5a^3 \times 3a^3 \). The coefficients here are 0.5 and 3. First, multiply these together:

• \( 0.5 \times 3 = 1.5 \)

By multiplying the coefficients, you simplify the numerical part of the expression, making it easier to combine with the variables.

When you multiply variables with the same base, you add their exponents. The base is the variable part of the term (in this case, 'a'). This rule is useful when simplifying expressions with multiplication.

In the example \( 0.5a^3 \times 3a^3 \), each term has the same base 'a'. So, according to the rule, you add the exponents:

• \( a^3 \times a^3 = a^{3+3} = a^6 \)

This simplifies the variables part of the expression by combining the exponents, resulting in a single term with a higher exponent.

After handling the coefficients and the exponents, the final step in simplifying an algebraic expression is to combine these results. This makes the expression more concise and easier to work with in further calculations.

For our example, after multiplying the coefficients (\( 1.5 \)) and adding the exponents of the variables (\( a^6 \)), we combine them together:

• \( 0.5a^3 \times 3a^3 = 1.5a^6 \)

This combination of the simplified coefficients and variables provides the final, simplified expression.

So, \( 0.5a^3 \times 3a^3 \) becomes \( 1.5a^6 \) after simplification. This process of algebraic simplification helps make complex expressions more manageable and can be applied to a wide range of algebraic problems.