Problem 31 - Algebra 1

Question

Simplify the expression. The simplified expression should have no negative exponents. $$\left(\frac{3}{x}\right)^{4}$$

Step-by-Step Solution

Answer

The simplified expression of \(\left(\frac{3}{x}\right)^{4}\) is \(81/x^4\).

Step 1: Understanding the given expression

Looking at the expression \(\left(\frac{3}{x}\right)^{4}\), we can see that it's actually \(3^4\) divided by \(x^4\). This comes from the rule that \( (a/b)^n = a^n / b^n \).

Step 2: Solving the individual exponents

Now, solve each of these individually. We have \(3^4\) equals 81, and \(x^4\) remains \(x^4\) as we don't know the value of x.

Step 3: Putting it together

Our simplified expression will be \(81/x^4\).

Key Concepts

Exponent Rules

Exponents play a crucial role in algebra, helping to succinctly express repeated multiplication. Mastering exponent rules can save time and simplify complex expressions. The Power of a Quotient Rule, for instance, signifies that when a fraction is raised to an exponent, we apply the exponent to both the numerator and the denominator independently. As such, \( (a/b)^n = a^n / b^n \). This is how we deduced that \(\left(\frac{3}{x}\right)^{4} = 3^4 / x^4 \).

There are several other rules like the Product of Powers Rule that states \( a^m \times a^n = a^{m+n} \), or the Power of a Power Rule indicating \( (a^m)^n = a^{m \cdot n} \). Knowing these rules enables students to combine terms with the same base and manage complex expressions efficiently. Remembering and applying these rules is an invaluable skill in algebra and beyond.

Simplify Algebraic Expressions

The aim of simplification is to make algebraic expressions as uncomplicated and readable as possible, often to facilitate further operations. Take the expression \(\left(\frac{3}{x}\right)^{4}\) for example. While it looks compact, it isn't fully simplified because it contains an exponent.

By applying exponent rules, we transform the expression into a more expanded form such as 81 over \(x^4\), which is considerably easier to work with, especially when dealing with equations. A simplification such as this could be the difference between quickly solving a problem and getting lost in algebraic manipulation. Always target to remove parentheses, consolidate like terms, and reduce fractions if possible. This helps in revealing the structure of the expression and makes future calculations smoother.

Negative Exponents

A negative exponent signifies the reciprocal of the base raised to the opposite positive exponent. For example, \( a^{-n} = \frac{1}{a^n} \). This means that any number with a negative exponent can be changed to a positive exponent by placing it under 1. If we had a negative exponent in the exercise \(\left(\frac{3}{x}\right)^{-4}\), we'd have to flip the fraction to get rid of them, resulting in \( x^4 / 3^4 \).

Understanding how negative exponents work is important not only to simplify expressions correctly but also to ensure that all final answers follow the mathematical convention of not having negative exponents unless specified.

Algebraic Operations

Algebra is full of operations—addition, subtraction, multiplication, division, and exponentiation being the primary ones. Mastery of these operations, especially exponentiation, is a must. In our example, exponents substantially changed the appearance and form of the expression \(\left(\frac{3}{x}\right)^{4}\).

When combining these operations, always follow the order of operations—parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). This structure ensures that every algebraic expression is simplified in a standard way, allowing for consistent results across different problems and for different individuals who may attempt to solve them.