Problem 13 - Algebra 1: Concepts and Skills

Question

Use the power of a quotient property to simplify the expression. $$ \left(\frac{-4}{3}\right)^{4} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(\left(\frac{-4}{3}\right)^{4}\) is \(\frac{256}{81}\)

Step 1: Power of Quotient Rule Application

Start by applying the power of a quotient rule to the expression \(\left(\frac{-4}{3}\right)^{4}\). According to the rule, a fraction to a power can be rewritten as the powers of the numerator and denominator separately which gives \(\frac{(-4)^{4}}{(3)^{4}}\)

Step 2: Evaluation of Powers

The next step is to calculate the numerators and denominators that are now exponents separately. For the numerator, (-4)^4, the negative sign is unnecessary as any real number to an even power results in a non-negative result. That is, \((-4)^{4} = 256\). For denominator, \(3^{4} = 81\)

Step 3: Simplified Expression

Substitute these back into the fraction to obtain the simplified expression as \(\frac{256}{81}\).

Key Concepts

Simplifying Expressions

Simplifying expressions involves transforming a complicated expression into a more manageable form without changing its value. In this exercise, we looked at the expression \(\left(\frac{-4}{3}\right)^{4}\). To simplify it, we used exponent rules and focused on breaking down each part of the expression. Understanding the structure of the expression is vital. We first rewrote the expression using the power of a quotient rule. This helps separate the expression into the powers of the numerator and the denominator.

By calculating these powers separately, you make the problem easier to manage. Evaluating \((-4)^4\) and \(3^4\) separately gives you a clearer path. At the end, putting these results back into one fraction simplifies our expression into \(\frac{256}{81}\). This process is essential and applies frequently in algebra, helping you streamline expressions efficiently.

Exponent Rules

Exponent rules are key to mastering expressions like \(\left(\frac{-4}{3}\right)^{4}\). Let's explore one of these rules used in this exercise called the "power of a quotient" rule. It states that a fraction raised to an exponent can be expressed as the numerator to the power and the denominator to the same power.

In simple terms, when you see \(\left(\frac{a}{b}\right)^{n}\), it transforms into \(\frac{a^n}{b^n}\). This rule helps break complex fractions into individual parts, allowing you to focus and simplify them separately.

In addition, knowing other basic exponent rules is crucial, such as:

• Multiplying like bases: \(a^m \times a^n = a^{m+n}\)
• Dividing like bases: \(\frac{a^m}{a^n} = a^{m-n}\)
• Anything to the power of zero is one: \(a^0 = 1\), where \(a e 0\)

These rules offer a foundation for understanding more complex expressions and simplifying them with confidence.

Fractional Exponents

Fractional exponents, while not directly used in the original exercise, play a significant role in mathematical expressions and simplification steps. A fractional exponent, such as \(a^{m/n}\), represents a power and a root operation. It is equivalent to \(\sqrt[n]{a^m}\), meaning you take the nth root of \(a\) and then raise it to the power of \(m\).

Understanding fractional exponents can transform seemingly complex equations into logical steps, much like handling whole number exponents. Here are some simple insights:

• The expression \(a^{1/n}\) is the same as the nth root of \(a\).
• When combining roots and powers, you can rewrite \(a^{m/n}\) as \((a^m)^{1/n}\) or \((a^{1/n})^m\), providing flexibility based on what's easier to compute.

Comprehending fractional exponents expands your toolkit for tackling expressions, making simplification smoother in varied algebraic contexts.