Problem 95 - College Algebra Essentials

Question

In Exercises \(91-100,\) simplify using properties of exponents. $$\left(x^{\frac{2}{3}}\right)^{3}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(\left(x^{\frac{2}{3}}\right)^{3}\) is \(x^2\).

Step 1: Identify the base and the exponents

In the expression \(\left(x^{\frac{2}{3}}\right)^{3}\), the base is \(x\) and the exponents are \(\frac{2}{3}\) and \(3\).

Step 2: Use the Property of Exponents

Now, use the property of exponents \((a^m)^n = a^{mn}\) to simplify. Multiply the exponents together to get the simplified form of the expression.

Step 3: Simplify the Exponents

Multiply the exponents \(\frac{2}{3}\) and \(3\) together: \(\frac{2}{3} * 3 = 2\). Therefore, \(\left(x^{\frac{2}{3}}\right)^{3} = x^2\).

Key Concepts

Exponent Simplification

When working with exponential expressions, one often encounters the task of exponent simplification. As seen in the exercise \(x^{\frac{2}{3}})^3\), where we have a power raised to another power, simplification involves using the rule \( (a^m)^n = a^{mn} \). This property allows us to multiply the exponents directly.

In our given problem, applying this rule is the key step. Since \(\frac{2}{3}\) and 3 are multiplied together, as per the rule, the result is 2. Thus, we achieve a simplified exponent: \( x^2 \). Simplifying exponents streamlines expressions and can make further algebraic operations considerably more manageable. It's crucial to remember that this rule applies only when the bases are the same and only when we have a power to a power situation.

Exponent simplification also encompasses other rules such as the product of powers rule \(a^m * a^n = a^{m+n}\), quotient of powers rule \(\frac{a^m}{a^n} = a^{m-n}\), and the power of a product rule \( (ab)^n = a^n * b^n \). Understanding and correctly applying these properties form the basis for working with exponent simplification.

Exponential Expressions

Exponential expressions, such as \(x^{\frac{2}{3}})^3\), play a significant role in algebra. They consist of a base, in this case, \(x\), and an exponent, which is \(\frac{2}{3}\) in the original expression before the simplification process.

An exponential expression represents repeated multiplication of the base. So, \(x^{\frac{2}{3}})^3\) initially implies that \(x\) is being multiplied by itself \(\frac{2}{3}\) times and then this result is raised to the power of 3.

Why does this matter?

Addressing this concept helps in other areas of mathematics, including compound interest calculations in finance, exponential growth in biology, and radioactive decay in physics.

In algebra, understanding how to manipulate these expressions is key to solving equations and simplifying complex algebraic expressions. Recognizing the different rules that apply to exponents unlocks the ability to handle a wide range of problems efficiently.

Algebraic Expressions

Algebraic expressions, such as the simplified result of our exercise, \(x^2\), consist of variables, numbers, and arithmetic operations. In this context, \(x\) is the variable which can represent any number, whereas \(2\) is the exponent indicating that \(x\) is squared, or multiplied by itself once.

Through algebraic expressions, we can represent real-world situations and solve problems by substituting variables with specific values. One of the benefits of simplifying expressions, as with exponent simplification, is to make it easier to evaluate, whether for specific values of \(x\) or in further algebraic manipulations.

Crucial Building Blocks:

Consider algebraic expressions as the building blocks for more complex mathematical concepts, including functions and polynomials. The ability to manipulate these expressions deftly is a fundamental skill in algebra that will be applied in almost every topic thereafter.

A solid grasp of algebraic expressions and their simplification is also essential for solving equations. It aids in recognizing patterns, predicting outcomes, and understanding the relationships between different elements within a problem.