Question

Write each expression without parentheses. Assume all variables are positive. $$ 3\left(10 e^{3 t}\right)^{2} $$

Short answer

Question: Simplify the given expression: $$ 3\left(10 e^{3 t}\right)^{2} $$ Answer: The simplified expression is $$300e^{6t}$$.

Step-by-step solution

Identify the expression

We are given the expression: $$ 3\left(10 e^{3 t}\right)^{2} $$ We need to simplify this expression by removing the parentheses.

Apply the power rule

We notice that our expression inside the parentheses is raised to the power of 2. We can apply the power rule: \((ab)^n=a^n b^n\). So, our expression becomes: $$ 3 \cdot \left(10^{2}\right) \cdot \left(e^{3 t}\right)^{2} $$

Simplify the expression inside the parentheses

Now we can simplify the expression inside the parentheses. We get: $$ 3 \cdot 100 \cdot e^{6 t} $$

Finish simplifying the expression

Now we can simplify the expression by multiplying 3 and 100 together: $$ 300 \cdot e^{6 t} $$ So, the simplified expression without parentheses is: $$ 300e^{6t} $$

Key concepts

Power Rule

Understanding the Power Rule is essential when working with exponents and expressions. The Power Rule helps simplify expressions that involve powers raised to another power. It states:

• When you have a product inside parentheses raised to an exponent, the exponent applies to each factor inside the parentheses separately.
• In mathematical terms, this is expressed as \[ (ab)^n = a^n imes b^n \].

Let's connect this rule with the problem at hand. We have the expression \(3(10e^{3t})^2\)\. According to the Power Rule, the 2 exponent applies to both 10 and \(e^{3t}\) inside the parentheses. This results in \(3 \times (10^2) \times (e^{3t})^2\)\. By utilizing this rule, we effectively break down the expression and make it more manageable for further simplification. Remember, mastering the Power Rule paves the way for handling more complex expressions with ease!

Simplifying Expressions

Once we've applied the Power Rule, the next step is all about simplifying expressions. Simplification involves reducing expressions to their most concise form while preserving the original value.
After applying the Power Rule to our problem, we arrive at \(3 \times 100 \times e^{6t}\)\. Here, each component is simplified separately:

• First, compute \(10^2\), which simplifies to 100.
• Next, handle the expression \( (e^{3t})^2 = e^{6t}\), by using the rule \( (a^m)^n = a^{mn} \).
• Finally, multiply 3 by 100 to get 300, resulting in the final simplified form \(300e^{6t}\).

Simplifying complex expressions might seem challenging at first, but breaking them into smaller pieces and applying rules step-by-step makes the process much easier.

Algebraic Expressions

Algebraic expressions form the backbone of algebra, consisting of variables, numbers, and operations. In our exercise, expression \(3(10e^{3t})^2\) combines multiplication and exponentiation—all common operations in algebraic expressions.
To simplify it, understanding each element is crucial:

• Variables like \(e^{3t}\) represent unknown quantities or can denote specific values, depending on the context.
• Numbers, such as 3 and 10, are constants that provide the expression's fixed components.
• Operations like multiplication and exponentiation indicate the math actions needed.

A deeper grasp of algebraic expressions allows us to manipulate and solve them confidently. By applying rules like the Power Rule and simplifying each component, we can transform complex algebraic expressions into simpler, more comprehensible forms, aiding in problem-solving across various mathematical fields.