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Cubing a number might sound complex at first, but it's simply a matter of multiplying that number by itself two additional times. In other words, if we say to cube the number 4, what we actually mean is to calculate the product of 4 times 4 times 4, which is 64. This operation is represented mathematically as 4 to the power of 3, written as \( 4^3 \). The cube of a variable, such as \( x \), is similarly noted as \( x^3 \).

When an algebraic term involves both a coefficient (a number) and a variable, like \( 4x \), the process is the same, although additional attention must be paid to the multiplication rules for both constants and variables. Therefore, cubing \( 4x \) means calculating \( (4x) * (4x) * (4x) \), which results in 64 for the numerical part and \( x^3 \) for the variable part.

Multiplying algebraic terms requires an understanding of both exponent rules and basic multiplication. To multiply terms with the same base, we simply add their exponents. For instance, when multiplying \( x \) by itself, \( x * x \), the result is \( x^2 \) because the exponents are each 1 (implied when an exponent is not written), and 1+1 equals 2.

When dealing with a coefficient, such as in \( 4x \), we multiply the coefficients normally and follow the rule about adding exponents for the like bases. For example, the calculation \( 4x * 4x * 4x \) involves multiplying the numeric coefficients (4*4*4) and then adding the exponents of the variable part (since \( x \) by default has an exponent of 1, it becomes \( x^3 \) when cubed). Thus, we end up with \( 64x^3 \) as the resulting product.

Expanded form breaks down expressions so that all operations are visible. When expressing powers, such as cubes, in expanded form, we show the multiplication step by step. If we take the cubed term \( 64x^3 \), the expanded form would literally demonstrate the term multiplied by itself twice, as \( 64 * x * x * x \).

In educational contexts, expanding powers helps to understand how exponents consolidate repeated multiplication. It makes the process transparent and can be especially clarifying when dealing with variables and coefficients together. While the exponent tells us how many times the base is used as a factor in the multiplication, the expanded form provides the meticulous detail of that process, which can aid in mastering the concept of exponents in algebra.