Problem 86 - Algebra A Combined Function

Question

Simplify each expression. $$ \left(t^{5}\right)^{11} $$

Step-by-Step Solution

Answer

The simplified expression is \( t^{55} \).

Step 1: Identify the Power of a Power Rule

The expression \( (t^5)^{11} \) involves an exponentiation of another exponent, which falls under the 'power of a power' rule in exponents. This rule states that \( (a^m)^n = a^{m \times n} \).

Step 2: Apply the Power of a Power Rule

Use the 'power of a power' rule to rewrite the expression by multiplying the exponents: \( t^{5 \times 11} \).

Step 3: Multiply the Exponents

Compute the multiplication: \( 5 \times 11 = 55 \). So the expression becomes \( t^{55} \).

Step 4: Write the Simplified Expression

The simplified expression for \( (t^5)^{11} \) is \( t^{55} \).

Key Concepts

Power of a Power Rule

The power of a power rule is a fundamental concept in exponentiation that helps simplify expressions involving exponents. It comes into play when you have a power raised to another power, like \((a^m)^n\). This scenario means you have a base, \(a\), that is raised to the power of \(m\), and then this entire expression is raised again to the power of \(n\).

Utilizing the rule is quite simple:

• You multiply the exponents in the expression.
• The rule can be mathematically expressed as \((a^m)^n = a^{m \times n}\).

Let’s say you have an expression \((t^5)^{11}\). Here, the base \(t\) is first raised to the power of \(5\) and then again to the power \(11\). According to the power of a power rule, you multiply these exponents (5 and 11) to simplify the expression.

Applying the rule gives you \(t^{5 \times 11} = t^{55}\). This powerful tool helps in reducing complex exponent problems into simple calculations.

Simplifying Expressions

Simplifying expressions involves breaking down complex mathematical expressions into their simplest form. This not only makes them easier to read and understand but also simplifies calculations. When dealing with exponents, simplifying often involves applying rules such as the power of a power rule.

To simplify an expression like \((t^5)^{11}\),

• Identify any applicable rules like the power of a power.
• Apply these rules step-by-step to achieve the most basic form possible.

In this case, by recognizing it is a 'power of a power' situation, we apply that specific rule. By doing so, we turn the expression \((t^5)^{11}\) into \(t^{55}\).

The simplified form \(t^{55}\) is much easier to comprehend and handle in further calculations or problem-solving tasks. Simplifying expressions also aids in avoiding mistakes by making them more straightforward.

Multiplying Exponents

When it comes to multiplying exponents, understanding the context is crucial. In mathematics, multiplying exponents arises when you deal with the power of a power. Although at first glance it might seem related to adding or multiplying numbers, it follows its own unique rule set.

Essentially, you multiply exponents when two or more exponents are related to each other through additional layers of exponentiation.

• Consider \((t^5)^{11}\) as an example. You have two exponents: 5 and 11.
• The task involves multiplying these exponents as part of applying the power of a power rule.

By computing this, \(5 \times 11\), the expression turns into \(t^{55}\). The process involves straightforward multiplication, turning potentially complex expressions into manageable forms.

Remember, whenever you deal with nested powers in mathematics, always check if the multiplying exponents rule from the power of a power can be applied.