Problem 77 - College Algebra with Modeling and Visualization

Question

Simplify the expression and write it with rational exponents. Assume that all variables are positive. $$ \left(x^{2}\right)^{3 / 2} $$

Step-by-Step Solution

Answer

The simplified expression is \(x^3\).

Step 1: Understand the Given Expression

The given expression is \( \left(x^{2}\right)^{3/2} \). This expression involves an exponent raised to another exponent.

Step 2: Apply the Power of a Power Rule

The power of a power rule states \((a^m)^n = a^{m \cdot n}\). Applying this rule, we have: \(\left(x^{2}\right)^{3/2} = x^{2 \cdot 3/2}.\)

Step 3: Simplify the Exponents

Now, simplify the exponent by performing the multiplication: \(2 \cdot \frac{3}{2} = 3\). Thus, \(x^{2 \cdot 3/2} = x^3\).

Step 4: Present the Final Expression

The simplified expression with a rational exponent is \(x^3\).

Key Concepts

Power of a Power Rule

When working with exponents, there is a helpful rule called the **Power of a Power Rule**. This rule simplifies expressions where an exponent is raised to another exponent.

The general form of the rule is \((a^m)^n = a^{m \cdot n}\). What this means is, when you have a power inside parentheses and another power outside, you multiply the exponents together. This way, you can unroll complicated exponent expressions.

For example, given the expression \(\left(x^{2}\right)^{3/2}\), utilize the Power of a Power Rule by multiplying the exponents, turning it into \(x^{2 \cdot 3/2}\). This operation makes it significantly simpler to work with. Understanding and applying this rule saves a lot of time when simplifying complex expressions that involve exponents.

Exponentiation

**Exponentiation** is a fundamental mathematical operation involving numbers, denoting repeated multiplication of a base number by itself. It's represented by the symbol "exponent". For instance, \(a^b\) means that \(a\) is multiplied by itself \(b\) times.

Exponents can be whole numbers, fractions (rational exponents), or even negative numbers. Rational exponents, like \(3/2\) in \(x^{3/2}\), express roots and powers, capturing both operations compactly in a single number. Exponents reflect the power and efficiency of equations and solve large multiplications quickly.

In our example, exponentiation helps transform the complex expression \(\left(x^{2}\right)^{3/2}\) into the clean form \(x^3\), demonstrating its usefulness in simplifying equations and solving algebraic expressions efficiently.

Simplifying Expressions

**Simplifying Expressions** involves reducing a mathematical expression into its simplest form. This often requires the application of various mathematical rules and operations, including exponent laws.

The goal is to make expressions easier to understand and use without changing their value. A simplified expression is typically shorter and more elegant. In algebra, simplifying involves combining like terms and canceling unnecessary elements.

In the exercise, simplification uses the exponent rules to turn \(\left(x^2\right)^{3/2}\) directly into \(x^3\). By using multiplication, the more complex fraction-ridden exponent disappears, revealing a neat and straightforward result. Simplifying such expressions not only clarifies the outcome but also helps when inputs must be plugged back into other calculations, promoting efficiency in mathematical problem-solving.