Question

Write each expression without parentheses. Assume all variables are positive. $$ 3\left(2^{x} e^{x}\right)^{4} $$

Short answer

Question: Simplify the given expression: \(3\left(2^{x} e^{x}\right)^{4}\) Answer: The simplified expression is \(3\cdot 2^{4x}\cdot e^{4x}\).

Step-by-step solution

Identify the expression inside the parentheses

We are given the expression \(3\left(2^{x} e^{x}\right)^{4}\). The expression inside the parentheses is \(2^{x}e^{x}\).

Apply the exponent property

We will apply the exponent property \((ab)^n = a^n b^n\). In our case, we have the power of 4 to apply to both \(2^{x}\) and \(e^{x}\). $$ 3\left(2^{x} e^{x}\right)^{4} = 3\left( (2^{x})^{4}\right)\left( (e^{x})^{4}\right) $$

Simplify the expressions using the properties of exponents

For the next step, we apply the property of exponents \((a^{n})^m = a^{nm}\). So, we have: $$ 3\left( (2^{x})^{4}\right)\left( (e^{x})^{4}\right) = 3\cdot 2^{4x}\cdot e^{4x} $$

Write the final simplified expression without parentheses

Finally, the simplified expression without parentheses is: $$ 3\cdot 2^{4x}\cdot e^{4x} $$ Thus, the given expression, \(3\left(2^{x} e^{x}\right)^{4}\), can be rewritten as \(3\cdot 2^{4x}\cdot e^{4x}\) without parentheses.

Key concepts

Properties of Exponents

Exponents are a powerful tool in mathematics, especially when working with algebraic expressions. Understanding their properties allows us to simplify complex expressions into more manageable forms.
When we talk about the "properties of exponents," we're referring to the rules that govern how exponents can be manipulated. One important property is

• **Power of a Product Property**: If you have an expression like \((ab)^n\), this can be expanded as \(a^n \, b^n\).

For instance, if you have \( (2^{x} e^{x})^{4} \), using the power of a product property becomes essential.

• **Power of a Power Property**: When you have an exponent applied to an exponent, such as \((a^n)^m\), it simplifies to \(a^{nm}\).

These properties of exponents make it much easier to work with and simplify algebraic expressions.

Simplifying Expressions

Simplifying expressions involves using mathematical operations and properties to reduce an expression to its simplest form. This process often involves removing parentheses and combining like terms.
In our exercise, the goal is to express \(3(2^{x} e^{x})^4\) without parentheses, which involves several steps:

• First, break down the expression using the properties of exponents, specifically the power of a product property, \((ab)^n = a^n b^n\). This allows each element inside the parentheses to be raised to the power separately.


• Next, apply the power of a power property,\((a^{n})^m = a^{nm}\). Each base inside the parentheses is raised to the expanded power.

By applying these steps, we transform the expression into \(3 \cdot 2^{4x} \cdot e^{4x}\). Each simplification step creates an expression that is easier to read and work with.

Algebra

Algebra is an essential field of mathematics focused on the study of symbols and the rules for manipulating these symbols. It forms a bridge between arithmetic and advanced mathematics, allowing us to express generalizations and relationships between numbers.
One of the key skills in algebra is simplifying expressions using the properties of exponents, as seen in this exercise. Expression simplification involves identifying terms that can be combined or manipulated according to mathematical properties.

• Expressions often include variables, which are symbols (like \(x\)) that stand in for numbers in equations and formulas.
• Algebraic manipulation helps solve equations by isolating variables and simplifying terms.

In this instance:

• We start with an algebraic expression that contains exponents and use algebraic techniques to simplify it, bringing clarity and ease to its manipulation.
• By understanding the properties of exponents and practicing simplifying expressions, students gain confidence in handling a variety of algebraic situations.

Mastering these core concepts is key to advancing in algebra and applying it successfully in more complex mathematical problems.