Problem 62 - Algebra 1: Concepts and Skills

Question

Write the expression as a single power of the base. $$ t \cdot\left(t^{3}\right) $$

Step-by-Step Solution

Answer

The expression \( t \cdot t^3 \) simplifies to \( t^4 \) when the multiplication exponent rule is applied.

Step 1: Identify the base and the exponents

The base in this expression is \( t \). It appears twice, once with an implicit exponent of 1 (since \( t = t^1 \)) and once with an explicit exponent of 3 in the expression \( t^3 \).

Step 2: Apply the multiplication exponent rule

When multiplying two expressions with the same base, according to the multiplication exponent rule, the exponents should be added up. Consequently, the expression can be simplified as follows: \( t \cdot t^3 = t^{(1+3)} \).

Step 3: Simplify the expression

The expression \( t^{(1+3)} \) simplifies to \( t^4 \) when the exponents are added up.

Key Concepts

Base

In mathematics, the term "base" refers to the number or variable that is repeatedly multiplied by itself according to the exponent's value. Understanding the base is fundamental when working with exponents as it acts like the foundation upon which the power is built.
For instance, in the expression \( t \cdot t^3 \), "\( t \)" is the base. It is the number being raised to a power, like "\( t \)" to the first power in the term \( t \), and "\( t \)" to the third power in the term \( t^3 \).

Here are some important points to remember about the base in exponentiation:

• It can be any number or variable.
• The base stays the same when applying exponent rules.
• Changing the base changes the entire expression.

Grasping the concept of a base helps in applying further rules like the multiplication exponent rule.

Multiplication Exponent Rule

The multiplication exponent rule is an efficient shortcut used when you are multiplying two powers that have the same base. According to this rule, you simply add the exponents together while keeping the base unchanged.
This rule can be illustrated using our example: \( t \cdot t^3 \).

Here is how you apply it:

• Identify the common base, which is \( t \).
• Add the exponents together: in this case, the implicit exponent is 1 (since \( t = t^1 \)). So, the calculation becomes: \( 1 + 3 \).
• Write the result with the same base and the new exponent: \( t^4 \).

This rule greatly simplifies calculations and expressions by reducing multiple terms into a single term with the summed exponent. It's essential to handle the same base to employ this rule, as mixing different bases would not adhere to the multiplication exponent formula.

Simplifying Expressions

Simplifying expressions is a process of transforming a mathematical expression into its most compact form, where all possible operations have been completed and all like terms have been combined. This makes the expression easier to read and work with.

Let's take the expression \( t \cdot t^3 \). By applying the steps from the multiplication exponent rule, we have simplified it to \( t^4 \).
This involves several key points:

• First, identify any rules (like the multiplication exponent rule) that can be applied.
• Utilize those rules to combine terms efficiently.
• Check if there are further simplifications possible after initial reductions.
• Ensure that the simplified expression maintains the integrity and equality of the original expression.

When expressions are simplified, they become more accessible for further calculations, easier to compare, and more straightforward to interpret. Hence, mastering simplification techniques is highly beneficial for solving more complex algebraic problems effectively.