Question

Simplify the expression. Write your answer using only positive exponents. $$ \left(\frac{2}{3}\right)^4 $$

Short answer

The simplified expression, written using only positive exponents, is \(\frac{16}{81}\).

Step-by-step solution

Analyze the expression

We are given the expression \(\left(\frac{2}{3}\right)^4\). This indicates that the fraction \(\frac{2}{3}\) is to be raised to the power 4.

Apply the power of a fraction rule

The power of a fraction rule states that \((\frac{a}{b})^n = \frac{a^n}{b^n}\). Applying this rule to our expression, we have \(\left(\frac{2}{3}\right)^4 = \frac{2^4}{3^4}\).

Simplify the expression

We now calculate \(2^4\) and \(3^4\). Simplifying, we have \(\frac{2^4}{3^4} = \frac{16}{81}\).

Key concepts

Positive Exponents

Working with exponents is a fundamental skill in algebra, and understanding positive exponents is crucial. An exponent signifies how many times a number, known as the base, is multiplied by itself. When we talk about positive exponents, we're referring to a base raised to a power that is a positive whole number. For example, in the expression \(2^3\), the base is 2 and the exponent is 3, indicating that 2 must be multiplied by itself two additional times, for a result of 2 × 2 × 2 = 8.

Positive exponents follow specific laws that make computing them straightforward, such as the product of powers rule \( a^m \cdot a^n = a^{m+n} \) and the power of a power rule \( (a^m)^n = a^{m \cdot n}\). In our exercise, we see a positive exponent applied to a fraction, demonstrating how these laws operate in different contexts. It's essential to remember that when an exponent is positive, the value increases, compared with negative exponents which denote division, leading to a smaller value of the expression.

Power of a Fraction

Raising a fraction to a power is a common operation in algebra that can intimidate students at first glance. However, the rule for the power of a fraction simplifies the process. This rule dictates that when you have a fraction raised to an exponent, you apply the exponent to both the numerator and denominator separately. In Latex notation, this rule is expressed as \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\).

In essence, this means that the operation is distributive over the fraction. As we can see from the given exercise, \(\left(\frac{2}{3}\right)^4\) becomes \(\frac{2^4}{3^4}\) by applying the power to both 2 and 3 independently. An important aspect to remember is that this rule applies uniformly regardless of whether the fraction is proper, improper, or mixed; the exponent doesn't discriminate and affects the numerator and denominator just the same.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. When simplifying algebraic expressions, our goal is to reduce the expression to its simplest form, irrespective of whether it involves integers, fractions, or variable terms. Simplification often calls for operations like combining like terms or applying exponent rules, as seen in our example.

The process involves several steps that, when followed systematically, bring the expression to its simplest form. For instance, when we are given a power of a fraction, as in \(\left(\frac{2}{3}\right)^4\), we must first apply the power to both the numerator and the denominator before simplifying each component separately. The simplified result does not change the value of the expression; it merely presents it in a more understandable and manageable form. Simplifying algebraic expressions requires a firm grasp of arithmetic rules and properties of operations, a foundation that allows students to tackle more complex algebraic problems with confidence.