Problem 46 - Algebra A Combined Function

Question

Use the power rule and the power of a product or quotient rule to simplify each expression. $$ \left(\frac{q}{t}\right)^{11} $$

Step-by-Step Solution

Answer

\( \frac{q^{11}}{t^{11}} \)

Step 1: Apply the Power of a Quotient Rule

The Power of a Quotient Rule states that \( (\frac{a}{b})^n = \frac{a^n}{b^n} \). Apply this rule to the expression \( \left(\frac{q}{t}\right)^{11} \). This simplifies to \( \frac{q^{11}}{t^{11}} \).

Step 2: Simplify the Expression

There are no further simplifications or operations to perform on \( \frac{q^{11}}{t^{11}} \), as the expression is now in its simplest form.

Key Concepts

Power Rule

Exponents are a crucial part of mathematics, helping us express repeated multiplications in a compact form. The power rule for exponents greatly simplifies expressions involving exponents. The rule states that when you take a power to another power, you multiply the exponents. For example,

• If you have \, \( (x^m)^n \), then using the power rule, it becomes \, \( x^{m \times n} \).

Understanding this rule is essential when you are dealing with complex calculations. It helps reduce the number of steps required to solve an expression and prevents errors. However, in some problems, like in our exercise, it might not apply directly.
Still, knowing this rule supports your understanding of other related rules.

Power of a Quotient

The power of a quotient is a fundamental concept when dealing with fractions raised to an exponent. The rule defines how to handle division under a single exponent. It states that:

• The numerator and the denominator should be raised individually to the power.
• Mathematically, this rule is defined as \, \( (\frac{a}{b})^n = \frac{a^n}{b^n} \).

By using this rule, you can break down a potentially complicated expression into smaller parts, making calculations more manageable.
In our example, the expression \, \( (\frac{q}{t})^{11} \) simplifies to \, \( \frac{q^{11}}{t^{11}} \), which is much easier to handle.
Recognizing when to apply the power of a quotient is key to efficiently simplifying expressions.

Simplifying Expressions

Once you've applied all your rules with exponents, like the power of a quotient, next comes simplification. Simplifying an expression often means putting it in its most straightforward, readable form. For example:

• Reducing expressions to the lowest possible terms.
• Ensuring there are no unnecessary operations left.

In the exercise given, after applying the power of a quotient, you end up with \, \( \frac{q^{11}}{t^{11}} \). This expression is already simple since there are no further operations or terms to reduce.
Understanding simplification is more about recognizing when an expression really is as simple as it can be.
It saves time and confusion when dealing with more extensive problems later on.