Problem 101 - Algebra A Combined Function

Question

Fill in each box with the correct expression. $$ \square \cdot a^{2 / 3}=a^{3 / 3}, \text { or } a $$

Step-by-Step Solution

Answer

The expression in the box is \(a^{1/3}\).

Step 1: Understand the Problem

We are given an expression \(\square \cdot a^{2/3} = a\). We need to find the expression that would go in the \(\square\) to satisfy this equation. \(a\) can be expressed as \(a^{3/3}\).

Step 2: Identify the Missing Expression

Since \(a = a^{3/3}\), we need to find a number or expression \(x\) so that combining it with \(a^{2/3}\) results in \(a^{3/3}\). Therefore, \(x \cdot a^{2/3} = a^{3/3}\).

Step 3: Solve for the Expression

To solve for \(x\), we divide both sides of the equation \(x \cdot a^{2/3} = a^{3/3}\) by \(a^{2/3}\). Thus, \(x = \frac{a^{3/3}}{a^{2/3}} = a^{(3/3) - (2/3)} = a^{1/3}\).

Step 4: Verify the Solution

We verify our solution by checking if \(a^{1/3} \cdot a^{2/3} = a\). Add the exponents: \(a^{(1/3) + (2/3)} = a^{3/3} = a\), which confirms our solution is correct.

Key Concepts

Fractional Exponents

Fractional exponents, also known as rational exponents, are a way to represent powers and roots within a single mathematical notation. They provide a more universal way of expressing roots as exponents.
A fractional exponent is expressed as:

• \( a^{m/n} \)

Here, \(a\) is the base, \(m\) is the power, and \(n\) is the root. Think of \(m/n\) as having two roles: \(m\) tells you how many times to multiply the base, while \(n\) indicates the degree of the root.

• If \( n = 2 \), we are dealing with a square root: \( a^{1/2} \) is the same as \( \sqrt{a} \).
• Similarly, \( a^{1/3} \) corresponds to the cube root: \( \sqrt[3]{a} \).

In our exercise, \(a^{2/3}\) and \(a^{3/3}\) are simplifications using fractional exponents. Understanding how to manipulate these expressions is crucial for solving equations involving fractional exponents.

Expressions

Expressions in algebra are combinations of variables, numbers, and operations. They can be as simple as \(x + 3\) or more complex like \(3x^2 + 2x - 5\). In our exercise, expressions involve applying the rules of exponents to modify and solve equations.
The expression \(a^{2/3}\) involves a fractional exponent, where \(a\) is raised to the power of \(2/3\). Similarly, \(a^{3/3}\) simplifies to \(a\) because the exponent equals 1.
To manipulate these expressions, remember:

• Add exponents when multiplying the same base: \(a^m \cdot a^n = a^{m+n}\).
• Subtract exponents when dividing: \(\frac{a^m}{a^n} = a^{m-n}\).

Expressions with fractional exponents need care, but mastering them simplifies many algebra problems.

Equation Solving

Solving equations is about finding the value of unknown variables that make the equation true. You need to be methodical and use rules of algebra to isolate the variable and solve the equation step by step.
In the exercise, our task was to find the expression that, when multiplied by \(a^{2/3}\), equals \(a\).

• Start by recognizing \(a = a^{3/3}\), allowing you to work with uniform exponent forms.
• Write the equation: \(x \cdot a^{2/3} = a^{3/3}\) and solve for \(x\).
• Divide both sides by \(a^{2/3}\) to isolate \(x\) on one side: \(x = \frac{a^{3/3}}{a^{2/3}} = a^{1/3}\).

This method confirms that to solve for an expression involving exponents, rules of exponents are essential. Consistently apply these rules to simplify and solve equations effectively.