Problem 85 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 85-90: Replace the symbol \(\square\) with either \(=\) or \(\neq\) to make the resulting statement true, whenever the expression has meaning. Give a reason for your answer. $$ \left(a^{r}\right)^{2} \square a^{\left(r^{2}\right)} $$

Step-by-Step Solution

Answer

Replace \( \square \) with \( = \) if \( r = 0 \) or \( r = 2 \); otherwise use \( \neq \).

Step 1: Expand the Left Side

First, expand the left side of the equation \( \left(a^{r}\right)^{2} \) using the power of a power property which states that \( (x^m)^n = x^{m \cdot n} \). This gives us: \[ \left(a^{r}\right)^{2} = a^{r \times 2} = a^{2r}. \]

Step 2: Analyze the Right Side

The right side of the expression is \( a^{r^2} \). This remains in its exponentiated form and is not simplified further, giving us: \[ a^{r^2}. \]

Step 3: Compare Exponents

Now, compare the exponents from both sides of the expression: \( a^{2r} \) and \( a^{r^2} \). For these expressions to be equal, their exponents must be the same, hence: \[ 2r = r^2. \]

Step 4: Solve the Exponent Equation

Rearrange the equation to solve for \( r \): \[ r^2 - 2r = 0. \] Factor the equation: \[ r(r - 2) = 0. \] This gives two possible solutions for \( r \): \( r = 0 \) or \( r = 2 \).

Step 5: Determine Equality or Inequality

When \( r = 0 \) or \( r = 2 \), both sides of the equality become the same. For any other value of \( r \), \( a^{2r} eq a^{r^2} \) since \( 2r eq r^2 \) generally holds. **Conclusion**: \( \square \) should be \( = \) if \( r = 0 \) or \( r = 2 \), and \( eq \) otherwise.

Key Concepts

Exponentiation

In algebra, exponentiation is a mathematical operation involving two numbers, the base and the exponent. It describes repeated multiplication of a number by itself. For instance, the expression \(a^r\) denotes \(a\) raised to the power of \(r\). The exponent tells us how many times to multiply the base by itself.

When we multiply powers with the same base, we can use certain rules. One important rule is the power of a power property, which states that \((x^m)^n = x^{m \cdot n}\). This concept was used in the exercise to simplify the left-hand side: \( \left(a^{r}\right)^{2} = a^{r \times 2} = a^{2r}\).

This simplification helps in comparing expressions by reducing complexity. It's crucial to always match powers properly to maintain equality. If the bases are the same, the focus shifts to their exponents when solving or proving equations.

Equation Solving

Equation solving is the process of finding the values for variables that make an equation true. In the context of algebra, it often involves transforming an equation until a solution becomes clear. Using properties of exponents, we derived the equation \( 2r = r^2 \) from comparing powers on both sides of the simplification.

To solve this, first set it to zero: \( r^2 - 2r = 0 \). This can be solved by factoring, resulting in \( r(r - 2) = 0 \). Solve it by finding values of \( r \) that satisfy the equation:

• \(r = 0\)
• \(r = 2\)

Both solutions make the original statement true when substituted back into the expression.

This approach underscores the importance of transforming equations carefully and systematically to reveal possible solutions. Always verify solutions by substituting them back in.

Inequality Analysis

Inequality analysis involves understanding when two expressions are not equal. In our exercise, we determined the conditions for equality. Outside those conditions, the expressions are unequal, or \(eq\).

When analyzing the inequality \( a^{2r} eq a^{r^2} \), it's essential to understand that the equality \( 2r = r^2 \) holds only when \(r\) is \(0\) or \(2\). For any other value of \(r\), the equation does not balance:

• For \(r = 1\), for example, \(2 \times 1 = 2\) does not equal \(1^2 = 1\).
• This illustrates how specific solutions are rare among many possibilities.

The inequality analysis provides a framework to reason about expressions and determine domains where true statements hold. It also helps in identifying boundaries, so we understand what conditions lead to inequality clearly.