Problem 123 - Algebra A Combined Function

Question

Fill in the boxes so that each statement is true. (More than one answer may be possible for these exercises.) $$ x^{\square}=\frac{1}{x^{5}} $$

Step-by-Step Solution

Answer

The exponent is -5.

Step 1: Understand the problem

We have an equation involving an exponent, specifically the expression \( x^{\square}=\frac{1}{x^{5}} \). We need to find the exponent that makes this statement true.

Step 2: Rewrite the expression

The expression \( \frac{1}{x^{5}} \) can be rewritten in terms of negative exponents. Recall that \( \frac{1}{x^n}=x^{-n} \). Using this, we can rewrite \( \frac{1}{x^{5}} \) as \( x^{-5} \).

Step 3: Equate the exponents

Since \( x^{\square} \) is equal to \( x^{-5} \), the exponent needed is the one that equals \(-5\). Thus, \( \square = -5 \).

Step 4: Verify the solution

Substitute \( -5 \) back into the original expression to verify. If \( \square = -5 \), then \( x^{-5} = \frac{1}{x^{5}} \), which is true. The expression is satisfied and correctly rewritten.

Key Concepts

Negative Exponents

Negative exponents can initially be a bit confusing, but they are simply another way to represent reciprocals. When you see a negative exponent, such as \( x^{-n} \), it indicates that the base \( x \) is on the bottom of a fraction. This means that \( x^{-n} \) is the same as \( \frac{1}{x^n} \).
For example:

• \( x^{-2} = \frac{1}{x^2} \)
• \( a^{-3} = \frac{1}{a^3} \)

Recognizing this can simplify many algebraic problems. By rewriting negative exponents as fractions, it becomes much easier to manipulate equations and solve for unknowns. Negative exponents are often seen in algebraic expressions where division needs to be simplified or when working with scientific notation.

Algebraic Expressions

Algebraic expressions consist of variables (like \( x \) or \( y \)) combined with numbers and arithmetic operations. They can represent complex equations or simple operations.
Let's break down its components:

• Variables: Symbols that can change value within an expression.
• Constants: Specific numbers that do not change.
• Coefficients: Numbers placed in front of variables, showing multiplication.
• Operations: Include addition, subtraction, multiplication, and division.

Understanding these components is vital. It helps to assemble, manipulate, and solve expressions or equations. Whether you're solving for an unknown or evaluating an expression, mastering these parts will make the process smoother.

Equation Solving

Solving equations is like assembling a puzzle where you're finding the missing piece that makes the whole picture fit together. Equation solving involves finding a value for the unknown that satisfies the given equation.
Here are some strategies:

• Isolating the Variable: Rearrange the equation so the unknown is on one side.
• Using Inverse Operations: Employ inverse mathematical operations to simplify both sides.
  For example, use addition to cancel out a subtraction.
• Checking Solutions: Substitute back into the original equation to confirm your solution is correct.

In the given exercise, identifying the exponent \( \, -5 \) that satisfies the equation by rewriting \( \frac{1}{x^5} \) as \( x^{-5} \) demonstrates how negative exponents play into solving algebraic equations. With practice, these methods become second nature, making equation solving an approachable and systematic task.