Problem 88 - Precalculus: Functions and Graphs

Question

Replace the symbol \(\square\) with elther \(=\) or \(\neq\) to make the resulting statement true, whenever the expression has meaning. Give a reason for your answer. \(\sqrt{a^{r}} \square(\sqrt{a})\)

Step-by-Step Solution

Answer

Replace with \( = \) if \( r = 1 \); otherwise, use \( \neq \).

Step 1: Understanding the Expression

The given expression is \( \sqrt{a^r} \square (\sqrt{a}) \). We need to determine whether to replace \( \square \) with \( = \) or \( eq \) under given conditions for \( a \) and \( r \). The expression involves comparing two square roots.

Step 2: Simplifying the Square Roots

To simplify \( \sqrt{a^r} \), note that \( \sqrt{a^r} = (a^r)^{1/2} = a^{r/2} \). Similarly, \( \sqrt{a} \) can be expressed as \( a^{1/2} \). The comparison becomes \( a^{r/2} \square a^{1/2} \).

Step 3: Setting Conditions for Equality

For \( a^{r/2} = a^{1/2} \) to hold, the exponents must be identical. Thus, we require \( r/2 = 1/2 \), which simplifies to \( r = 1 \). If \( r = 1 \), the expression is true under all conditions where both sides are defined.

Step 4: Setting Conditions for Inequality

If \( r eq 1 \), then \( a^{r/2} \) does not equal \( a^{1/2} \), making \( \sqrt{a^r} eq \sqrt{a} \). Here, inequality holds when \( r eq 1 \).

Step 5: Conclusion

The expression \( \sqrt{a^r} = \sqrt{a} \) is valid if and only if \( r = 1 \). If \( r eq 1 \), the expression becomes \( \sqrt{a^r} eq \sqrt{a} \). Thus, the correct symbol depends on the value of \( r \).

Key Concepts

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 times 3 equals 9. When dealing with expressions involving square roots, such as \( \sqrt{a} \), our focus is on finding a number that, when squared, equals \( a \).

Square roots have particular rules that need to be followed:

• The square root of a product can be broken into the product of the square roots, \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \).
• The square root of a fraction is the fraction of the square roots, \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).
• Every positive number has two square roots: one positive and one negative, though in much mathematical practice, the positive root is assumed.

In our expression \( \sqrt{a^r} \), note that it simplifies to \( a^{r/2} \). Understanding these rules helps to make sense of expressions and simplifies solving them.

Exponents

Exponents, also known as powers, are a way to express repeated multiplication of a number by itself. For example, \( a^3 \) means \( a \times a \times a \). In general, \( a^n \) indicates that \( a \) is multiplied by itself \( n \) times.

When working with exponents, several important rules can be useful:

• Multiplying powers with the same base: \( a^m \cdot a^n = a^{m+n} \).
• Dividing powers with the same base: \( \frac{a^m}{a^n} = a^{m-n} \).
• Raising a power to another power: \( (a^m)^n = a^{m\cdot n} \).
• A power of zero: \( a^0 = 1 \) (as long as \( a eq 0 \)).

Understanding exponents is crucial when solving expressions like \( \sqrt{a^r} = a^{r/2} \). This particular expression shows that taking the square root of \( a^r \) is the same as raising \( a \) to the power of \( r/2 \). Whether solving for equality or inequality as in this context, these exponent rules simplify the calculations.

Inequality

An inequality is a mathematical sentence that shows the relationship between quantities that are not equivalent. For example, \( a < b \) indicates that \( a \) is less than \( b \).

There are several signs used in inequalities:

• \(<\): less than
• \(>\): greater than
• \(\leq\): less than or equal to
• \(\geq\): greater than or equal to
• \(eq\): not equal to

When comparing square roots like \( \sqrt{a^r} \) and \( \sqrt{a} \), recognizing that they are unequal when \( r eq 1 \) is key. When handling inequalities, it's important to remember that multiplying or dividing an inequality by a negative number reverses the inequality sign. Therefore, careful manipulation and an understanding of exponent and square root principles are essential in determining when inequalities hold.

Equation Solving

Solving equations involves finding the values of the variable that make the equation true. In our given expression \( \sqrt{a^r} \square \sqrt{a} \), determining the right symbol (either \( = \) or \( eq \)) necessitates solving for \( r \).

The steps to solve such an equation include:

• Identify the expression or equation.
• Simplify each side of the equation if necessary. With \( \sqrt{a^r} = a^{r/2} \), we simplify using exponent rules.
• Check if you can equate the expressions by finding a common base or power.
• Solve for the unknown by balancing the equation.

Specifically, in this exercise, solving the equality \( a^{r/2} = a^{1/2} \) leads to finding \( r = 1 \). Once you identify the conditions, you can set parameters for the solution, confirming when an equation holds true or when it diverges, aiding in applying the correct inequality or equality symbol in the process.