Problem 60 - Algebra 2

Question

Simplify. Assume that no variable equals \(0 .\) $$ 5 r t^{2}(2 r t)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(20r^3t^4\).

Step 1: Apply the Power Rule

The expression is given as \(5rt^2(2rt)^2\). Begin by simplifying \((2rt)^2\). According to the power rule \((ab)^n = a^n b^n\), expand this to \((2)^2 (r)^2 (t)^2\). This results in \(4r^2t^2\).

Step 2: Distribute the Inside Terms

Now, substitute the expanded form we obtained in Step 1 into the original expression: \(5rt^2 \times 4r^2t^2\). This becomes \(5rt^2 \, \times \, 4r^2t^2\).

Step 3: Multiply the Numeric Terms

Multiply the constant numeric parts: \(5 \times 4 = 20\). Keep this as the coefficient of the final expression.

Step 4: Combine the Variable Terms

Combine the variable terms by adding exponents with like bases. For \(r\), we have \(r^1 \times r^2 = r^{1+2} = r^3\). For \(t\), we have \(t^2 \times t^2 = t^{2+2} = t^4\).

Step 5: Write the Simplified Expression

The simplified expression, combining the results from the previous steps, is \(20r^3t^4\).

Key Concepts

Power Rule

The power rule is a fundamental rule in algebra that helps us manage expressions with exponents. It states that when you raise a power to a new exponent, you multiply the exponents. In the context of our problem, we are dealing with the expression \((2rt)^2\). Applying the power rule involves distributing the outer exponent to each element inside the parentheses:

• This means \((ab)^n = a^n b^n\).
• Thus, \((2rt)^2 = (2)^2 (r)^2 (t)^2\).

This simplification step helps break down the expression into smaller parts: \(4r^2t^2\). Each individual component \((2, r, t)\) is raised to the power of 2. This makes calculations simpler in subsequent steps, because we can easily handle each part separately.

Exponent Rules

Exponent rules are crucial for simplifying algebraic expressions with powers. When you multiply or divide terms with the same base, there are specific rules that apply:

• Multiplying like bases: Add the exponents. For example, \(r^1 \times r^2 = r^{1+2} = r^3\).
• Dividing like bases: Subtract the exponents.

In the given problem, we multiplied like bases after simplifying with the power rule. For the variable \(t\), we had \(t^2 \times t^2 = t^{2+2} = t^4\). This step is easy once you know the rule: keep the base and adjust the power based on multiplication or division. Thus, exponent rules streamline polynomial simplification by managing powers correctly.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and operations. In our problem, the algebraic expression is represented as \(5rt^2 \times 4r^2t^2\). Simplifying this kind of expression requires handling both coefficients (numeric parts) and terms (variable parts):

• Coefficients: Simply multiply them together, \(5 \times 4 = 20\).
• Terms with variables: Combine using exponent rules to manage powers and ensure accurate coefficients of variable terms.

Proper manipulation and simplification lead to the final reduced form, which in this case is \(20r^3t^4\). Each element of the expression plays a role in the final result, so attention to detail is essential.

Variables Manipulation

Variables manipulation involves operations like addition, subtraction, multiplication, and division, as well as careful attention to variables' exponents. Simplifying algebraic expressions by manipulating variables requires understanding relationships and interactions:

• Combining Variable Terms: Add exponents for like bases, such as \(r\) and \(t\).
• Distribution: Carefully follow mathematical operations like multiplication among coefficients and variables.

In our exercise, after simplifying the power of each term, combining terms becomes easier. \(r^1\) and \(r^2\) add up to form \(r^3\). Similarly, \(t^2\) and \(t^2\) add to \(t^4\). Hence, through meticulous manipulation, we achieve the expression \(20r^3t^4\). This manipulation ensures that the expression is simpler and more straightforward.