Question

Simplify the expression. \(\left(4 x^{4}\right)^{3}\)

Short answer

The simplified expression is \(64x^{12}\)

Step-by-step solution

Identify the expression

The expression to simplify is \(\left(4 x^{4}\right)^{3}\).

Apply the exponent rule

The task is to simplify a power of power, if m and n are any numbers, \((a^{m})^{n}=a^{m*n}\). Therefore, we multiply the exponents. The base 4, raised to the power of 3, is \((4^{3})\). The variable \(x^{4}\) is raised to the power of 3 so it becomes \((x^{4*3})\).

Compute the multiplication

The calculation gives us \(64x^{12}\).

Key concepts

Simplifying Expressions

Simplifying algebraic expressions is a foundational skill in mathematics, which involves reducing expressions to their most basic form without changing their value. To do this effectively, one must understand the order of operations and the relevant algebraic properties, such as the distributive property, combining like terms, and the laws governing exponents.

When simplifying an expression like \(\left(4 x^{4}\right)^{3}\), the first step is to look for any obvious simplifications, such as multiplying coefficients or applying exponent rules. In this case, there is an opportunity to simplify the expression by dealing with the exponent applied to both the numerical coefficient (4) and the variable term (\(x^{4}\)).

By recognizing that each component within the parentheses is raised to the power of 3, we're setting up the stage for easier computations. The numerical part, 4, is simple to calculate, and as for the variable part with the exponent, we'll see it’s just as straightforward when applying the 'power of a power' rule, which will be discussed in the next section.

Power of a Power

Understanding the 'power of a power' rule is essential when dealing with exponential expressions. This rule states that when you have a power raised to another power, you multiply the exponents. The mathematical representation of this rule is \( (a^{m})^{n} = a^{m \cdot n} \).

Let's break down what this means in the context of our example \(\left(4 x^{4}\right)^{3}\). We are essentially taking the number 4, which is raised to the power of 1 (since any number without a written exponent has an implied exponent of 1), and raising it to the power of 3. Similarly, we take the variable term \(x^{4}\) and raise it to the power of 3. According to our 'power of a power' rule, the exponents are multiplied to yield \(4^{1 \cdot 3}\) and \(x^{4 \cdot 3}\).

This simplifies down to \(4^{3}\) and \(x^{12}\), which leads us to our next step in the simplification process—performing the actual multiplication of the exponents and solving for the numerical base.

Exponential Expressions

Exponential expressions involve numbers or variables raised to a power. These expressions follow specific rules that make it easier to work with powers and simplify calculations, such as the multiplication of exponents we've seen in the 'power of a power' rule.

In our example, once we've applied the rule and multiplied the exponents, we are left with \(4^{3}\) and \(x^{12}\). Evaluating the expression \(4^{3}\) simply means multiplying 4 by itself three times: \(4 \cdot 4 \cdot 4\), which results in 64. For the variable portion, since we're dealing with the same base (x), we keep that base and use the new exponent, resulting in \(x^{12}\).

Upon completing these steps, we combine the two components to get our fully simplified expression, 64\(x^{12}\). This final expression is much easier to use in further calculations or equations, due to its simplified form, which doesn’t change its inherent value but makes subsequent mathematical operations more straightforward.