Problem 103 - Algebra A Combined Function

Question

Fill in each box with the correct expression. $$ \frac{\square}{x^{-2 / 5}}=x^{3 / 5} $$

Step-by-Step Solution

Answer

The expression in the box is \( x^{1/5} \).

Step 1: Understand the Problem

We have the expression \( \frac{\square}{x^{-2/5}} = x^{3/5} \). Our goal is to find the value of the box (\( \square \)) so that both sides of the equation are equal.

Step 2: Isolate the Square Placeholder

To isolate the box, multiply both sides of the equation by \( x^{-2/5} \). This gives us: \[ \square = x^{3/5} \times x^{-2/5} \]

Step 3: Simplify Using the Laws of Exponents

Apply the laws of exponents: when multiplying two exponents with the same base, add the exponents. Thus, \[ x^{3/5} \times x^{-2/5} = x^{(3/5) + (-2/5)} = x^{1/5} \]

Step 4: Final Answer

The expression that should be in the box is \( x^{1/5} \). This means \( \square = x^{1/5} \).

Key Concepts

Multiplying Exponents

When you multiply exponents with the same base, you simply add the exponents together. This is a fundamental rule in the laws of exponents. For example, consider the expression \( x^{a} \times x^{b} \). To multiply these, you add the exponents \( a \) and \( b \) together, which gives you \( x^{a+b} \).

• This rule applies only when the bases are the same. For instance, if you have \( x^3 \times x^2 \), the result would be \( x^{3+2} = x^5 \).
• It is essential to remember this rule when dealing with exponential expressions. It simplifies calculations and can transform complex exponential expressions into more manageable forms.

In the problem we are solving here, \( x^{3/5} \times x^{-2/5} \) uses this rule. By adding the exponents, we get \( x^{(3/5) + (-2/5)} = x^{1/5} \). This makes multiplying exponents a straightforward process once you remember the simple rule of adding the exponents.

Simplifying Expressions

Simplifying expressions means breaking them down into their simplest form. This often involves applying various algebraic rules, including those for exponents. Simplifying can make the expression easier to understand or use in future calculations.

• When dealing with exponents, this frequently involves using the rules we've discussed: adding exponents during multiplication and subtracting exponents during division.
• Look for terms that can be combined or canceled out. For exponents, check if they share the same base and can be manipulated using addition or subtraction.
• Ensure all the terms are maneuvered correctly according to the laws of exponents to achieve the simplest form of the expression.

In the exercise, the original expression \( \frac{\square}{x^{-2/5}}=x^{3/5} \) was simplified by multiplying both sides by \( x^{-2/5} \). This application of exponent rules ensured the equation was easier to manage ultimately leading to an expression as simple as \( x^{1/5} \). This also shows the importance of simplification in maintaining a clear and effective algebraic solution.

Isolating Variables

Isolating variables involves arranging an equation such that a specific variable is by itself on one side of the equation. This process often helps in solving equations and finding the value of the unknown.

• To isolate a variable, you need to perform operations that will keep the equation balanced. Whatever you do to one side, you must do to the other.
• When the variable is part of a division or multiplication involving another term, you can multiply or divide both sides by the reciprocal term to isolate it.
• Isolation might often require several steps, employing inverse operations such as adding to counteract subtraction, or multiplying to counteract division.

In the problem presented, the goal was to isolate the unknown \( \square \). By multiplying both sides by \( x^{-2/5} \), the variable \( \square \) was isolated on one side, allowing the equation to be solved straightforwardly to find that \( \square = x^{1/5} \). This illustrates how isolating variables ties into simplifying algebraic problems effectively and efficiently.