Problem 32 - Elementary and Intermediate Algebra

Question

Use the product rule for exponents to simplify each expression. Write the results using exponents. $$ (t+1)^{5}(t+1)^{3} $$

Step-by-Step Solution

Answer

\((t+1)^8\)

Step 1: Identify the Base

In the expression \((t+1)^{5}(t+1)^{3}\), both terms have the same base, \((t+1)\). The base remains unchanged when using the product rule for exponents.

Step 2: Apply the Product Rule for Exponents

The product rule for exponents states that for any base \(a\), if you have \(a^m\cdot a^n\), then it simplifies to \(a^{m+n}\). Use this rule to simplify the expression: \[(t+1)^{5} \cdot (t+1)^{3} = (t+1)^{5+3}\]

Step 3: Simplify the Exponent

Now, add the exponents: \[5 + 3 = 8\] So, the expression simplifies to: \[(t+1)^{8}\]

Key Concepts

Base in Algebra

In algebra, the term 'base' refers to the main component of an expression with an exponent. It is the number or variable that is multiplied by itself. In the case of \((t+1)^5\cdot(t+1)^3\), the base is \((t+1)\). Both terms in this expression have the same base, which is essential when applying the product rule for exponents. This rule simplifies expressions where multiple exponential terms share the same base.
Understanding the base is crucial:

• It acts as the foundation of any exponential expression.
• It stays unchanged when using the product rule.

Recognizing the base helps in correctly applying rules like the product rule of exponents and simplifies expressions efficiently.

Simplifying Expressions

Simplifying expressions in algebra involves reducing them to their most concise form. This makes calculations easier and solutions clearer. The product rule for exponents is a key strategy in simplifying expressions. By recognizing common bases and combining their exponents, larger and more complex expressions can be reduced efficiently.
In expressions like \((t+1)^5\cdot(t+1)^3\), the simplification process is:

• Identify the common base, \((t+1)\).
• Use the product rule to combine exponents: \((t+1)^{5+3}\).
• Result: \((t+1)^8\) - a simplified version of the expression.

By following these steps, you can simplify expressions effectively, paving the way for easier problem-solving in algebra.

Exponents in Algebra

Exponents are a fundamental part of algebra. They indicate how many times a base is multiplied by itself. The expression \((t+1)^3\) means that \((t+1)\) is multiplied by itself three times. Understanding how exponents work and how to manipulate them helps solve many algebra problems.
The product rule for exponents comes into play when multiplying terms with the same base. The rule states:

• If you have \(a^m\cdot a^n\), it simplifies to \(a^{m+n}\).
• This simplifies and reduces expressions by combining the powers.
• The expression \((t+1)^5\cdot(t+1)^3\) simplifies using this rule to \((t+1)^{5+3} = (t+1)^8\).

By mastering the rules of exponents, like the product rule, you can handle and simplify complex algebraic expressions with ease.