Problem 80 - Elementary Algebra Within Reach

Question

In Exercises \(75-86\), simplify the expression. $$ \left(-2 t^{3}\right)\left(4 t^{2}\right) $$

Step-by-Step Solution

Answer

The simplified form of the expression \(\left(-2 t^{3}\right)\left(4 t^{2}\right)\) is \(-8t^{5}\).

Step 1: Identify the Factors

In the expression \(\left(-2 t^{3}\right)\left(4 t^{2}\right)\), the two factors are \(-2t^3\) and \(4t^2\). We need to multiply these together.

Step 2: Multiply the Coefficients

The coefficients of the two factors are -2 and 4. Multiplication of these two gives us -8.

Step 3: Use the Exponent Rule

There are \(t^{3}\) and \(t^{2}\) in the separate factors. When multiplying expressions with the same base, add the exponents. Here, both expressions have the base \(t\), and the exponents are 3 and 2. Therefore, applying the exponent rule, we have \(t^{3+2}\), which simplifies to \(t^{5}\).

Step 4: Combine the Coefficients and Exponents

The final step is to combine the results from steps 2 and 3 to obtain the simplified form of the expression. From step 2, we have -8 and, from step 3, we have \(t^{5}\). Therefore, the simplified form of the expression is \(-8t^{5}\).

Key Concepts

Exponent Rules

Understanding exponent rules helps simplify expressions with powers. When multiplying terms with the same base, we add their exponents. For example, in the expression \(-2t^3\) and \(4t^2\), the base is \(t\). Thus, to multiply, one must add the exponents \(3\) and \(2\). This results in a new exponent: \(t^{3+2} = t^{5}\). This rule is not just for numbers. It works for any quantity raised to a power. More formally, if you have \(a^m \times a^n\), the result is \(a^{m+n}\). Using this rule makes simplifying algebraic expressions a breeze. Remember this method to handle expressions with powers efficiently.

Multiplying Expressions

Multiplying algebraic expressions involves a few straightforward steps. Start with the coefficients, the numerical parts of the terms. In the given example, the coefficients are \(-2\) and \(4\). Multiply them to get \(-8\).

• Step 1: Focus on coefficients like \-2\ and \4\.
• Step 2: Multiply them as regular numbers.
• Step 3: Notice, \(-2 \times 4 = -8\).

Next, multiply the variable parts separately, using exponent rules. For terms \(t^3\) and \(t^2\), add the exponents to get \(t^{5}\). Multiplication of algebraic expressions usually involves multiplying both coefficients and variables separately like this. Combining these results completes the multiplication process.

Simplifying Expressions

Simplifying expressions is about making them easier to work with. The objective is a more compact version of the original equation. First, multiply all numerical coefficients together. Then, apply exponent rules to deal with variables that have the same base. In our case, the expression \((-2t^3)(4t^2)\) simplified to \(-8t^5\).

• Step 1: Combine coefficient: \(-2 \times 4 = -8\).
• Step 2: Apply exponent rule: Like bases, add exponents \(3 + 2 = 5\).
• Step 3: Merge both results: \(-8 \times t^5 = -8t^5\).

The key is to break down the expression into manageable parts. Simplification reduces complexity and prepares the expression for further operations, like solving or graphing.