Problem 23 - Algebra 1: Concepts and Skills

Question

Simplify the expression. If not possible, write already in simplest form. $$\frac{t^{4}}{t^{2}(t+2)}$$

Step-by-Step Solution

Answer

The simplified form of the given expression \(\frac{t^{4}}{t^{2}(t+2)}\) is \(t^2/(t+2)\).

Step 1: Identify Any Common Factors

Look at the expression \(\frac{t^{4}}{t^{2}(t+2)}\). Here, you can observe that both the numerator (t^4) and the denominator (t^2(t+2)) have a common factor, which is t^2.

Step 2: Apply Quotient Rule of Exponents

The quotient rule states that when you divide two quantities with the same base, you can subtract the exponent of the denominator from the exponent of the numerator. This works out to be t^{4-2} which is written as t^2. So, the given expression simplifies to \(t^2/(t+2)\). Note that you only subtract the exponents of the terms that are common in both the numerator and denominator.

Step 3: Final Answer

So, the expression \(\frac{t^{4}}{t^{2}(t+2)}\) simplifies to \(t^2/(t+2)\).

Key Concepts

Quotient Rule of Exponents

The Quotient Rule of Exponents is a helpful tool when simplifying algebraic expressions. It comes into play when you divide expressions that have the same base. The rule is quite simple: if you have an expression \[ \frac{a^m}{a^n} = a^{m-n} \]where "a" is the common base and "m" and "n" are exponents, you subtract the exponent in the denominator from the exponent in the numerator.

This rule is really powerful because it allows you to simplify complex fractions into more manageable forms. For example, in the given expression \( \frac{t^4}{t^2(t+2)} \), you initially observe that the base 't' has exponents both in the numerator (\(t^4\)) and in the denominator (\(t^2\)).

By applying the quotient rule, you subtract the exponents: \[ t^{4-2} = t^2 \]This simplifies part of the expression neatly, helping you move one step closer to simplifying the whole expression.

Common Factors

Understanding common factors is another essential part of simplifying algebraic expressions. A common factor is a factor that is shared between two or more terms.

In the mathematical expression \( \frac{t^4}{t^2(t+2)} \), both the numerator \(t^4\) and the denominator \(t^2(t+2)\) share a common factor of \(t^2\).

Recognizing this allows you to simplify the fraction. Here's how you can think about it:

• The numerator \(t^4\) can be broken down into \(t^2 \cdot t^2\).
• The denominator has \(t^2\) right next to the other factor \((t+2)\).

By canceling out the common \(t^2\) from both parts, you reduce the expression to \( \frac{t^2}{t+2} \). This cancellation is key to making the expression simpler and easier to interpret.

Simplifying Rational Expressions

Simplifying rational expressions is a crucial skill in algebra that involves reducing fractions to their simplest form. A rational expression typically looks like \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials. The goal is to make both numerator and denominator as simple as possible.

For the expression \( \frac{t^4}{t^2(t+2)} \), we begin by identifying any appropriate simplification techniques, such as using the quotient rule of exponents and finding common factors, as demonstrated in previous sections.

Once the common \(t^2\) is removed using both methods, we are left with \( \frac{t^2}{t+2} \), which cannot be further simplified easily since no further common factors exist between \(t^2\) and \(t+2\). The expression is now in its simplest state.

Learning how to simplify rational expressions will empower you to handle more complex algebraic problems with confidence, making such expressions more digestible and easier for further algebraic operations.