Question

Simplify. $$ (2 t)^{4} \cdot 3^{3} $$

Short answer

The simplified form of the given expression is \(432 \cdot t^4\).

Step-by-step solution

Apply the exponential rule

Start by applying the exponent of 4 to each component within the parenthesis: \((2t)^4 = 2^4 \cdot t^4 = 16 \cdot t^4\). So, the initial expression can be rewritten as \(16 \cdot t^4 \cdot 3^3\).

Evaluate the exponential expression

Evaluate \(3^3\) to simplify further. The value of \(3^3\) is 27. The expression can be further simplified to \(16 \cdot t^4 \cdot 27\).

Multiply constants together

Multiply together the constant multipliers, 16 and 27. This results to \(432 \cdot t^4\).

Key concepts

Exponential Rule

Understanding the exponential rule is fundamental when simplifying expressions that include powers. When an expression like \( (a \cdot b)^n \) is given, the exponential rule allows you to apply the exponent to each base inside the parentheses individually. When simplifying \( (2t)^4 \), for instance, you separate the bases 2 and t and raise each of them to the fourth power, resulting in \( 2^4 \cdot t^4 \). This results in a clearer expression, \( 16 \cdot t^4 \), where the constants have been evaluated, leaving a neat power of the variable to work with.

The exponential rule also reminds us that the operation inside the parentheses must be multiplication for the property to apply. If the terms within the parentheses were being added or subtracted, we could not apply the exponent to them individually without using binomial expansion or other methods.

Evaluating Exponents

Evaluating exponents is a primary skill in algebra that involves finding out what a number raised to a power is. When you see an expression like \( 3^3 \), you need to calculate it by multiplying the base, which is 3, by itself as many times as the exponent, which is 3. So, \( 3^3 \) would be \( 3 \times 3 \times 3 = 27 \).

When evaluating exponents, especially for constants, recognizing common powers can save time. For example, knowing offhand that \( 2^4 = 16 \) or \( 3^2 = 9 \) can expedite solving problems. In the exercise, after determining that \( 3^3 = 27 \), you directly simplify the expression by replacing the exponential form with its evaluated counterpart, giving you \( 16 \cdot t^4 \cdot 27 \) as the next step towards a solution.

Multiplying Constants

Multiplying constants refers to the process of multiplying numbers that do not change, as opposed to variables which can represent any number. In an expression like \( 16 \cdot t^4 \cdot 27 \), the numbers 16 and 27 are constants. To further simplify the expression, you need to multiply these constants together.

This process is straightforward - for our example, you would calculate \( 16 \times 27 \), which gives 432. The simplified expression then becomes \( 432 \cdot t^4 \). Multiplying constants effectively reduces the problem to a more manageable size and presents a tidier and more understandable expression for anyone evaluating the mathematical statement.