Problem 52 - Algebra Form and Function

Question

Without solving them, say whether the equations in Problems \(43-56\) have a positive solution \(x=a\) such that (i) \(a>1\) (ii) \(\quad a=1\) (iii) \(\quad 0

Step-by-Step Solution

Answer

Answer: The positive solution for this equation belongs to the category where \(a > 1\).

Step 1: Rewrite the equation in a more familiar form

We can rewrite the equation \(x^{-3} = \frac{1}{4}\) as \(\frac{1}{x^3} = \frac{1}{4}\). Now the equation looks more familiar and easier to handle.

Step 2: Analyze the conditions

Since we have an equation in the form of \(\frac{1}{x^3} = \frac{1}{4}\), let's see if we can find any positive solution that fits the given categories. (i) If \(a > 1\), then \(x^3 < 1\), which means \(x < 1\) for non-negative values of \(x\). There is a possibility that the solution exists in this category. (ii) If \(a = 1\), then \(x^3 = 1\), which implies \(x = 1\). But the given equation says that \(\frac{1}{x^3} = \frac{1}{4}\), so a solution with x = 1 does not satisfy the equation. (iii) If \(0 < a < 1\), then \(x^3 > 1\), which means \(x > 1\) for non-negative values of \(x\). There is a possibility that the solution exists in this category.

Step 3: Solve the equation

To find the value of \(x\), we can cross-multiply the equation \(\frac{1}{x^3} = \frac{1}{4}\), which gives us \(4 = x^3\). Now, solving for \(x\), we get \(x = \sqrt[3]{4}\). Since \(\sqrt[3]{4} > 1\), the solution falls into the category (i): \(a > 1\).

Step 4: Conclusion

The given equation \(x^{-3} = \frac{1}{4}\) has a positive solution \(x=a\) such that \(a>1\), as \(x = \sqrt[3]{4}\).

Key Concepts

Positive Solutions in Equations

Positive solutions in equations refer to finding values greater than zero that satisfy a given mathematical equation. In the context of algebra, these solutions are particularly interesting when dealing with specific conditions or intervals for the unknown variable, such as:

• Greater than 1: Here, the possible solutions are those that fall above 1.
• Exactly 1: The solution must exactly equal 1 if it exists.
• Between 0 and 1: Here, the solution lies strictly between zero and one.
• No positive solution: It indicates that the equation does not have any solution greater than zero.

These conditions help us classify or predict where the solution might fall without actually solving the equation. By understanding these classifications, we can often gain insights into the nature of the equations just by examining their forms. For instance, if an equation like \(x^{-3} = \frac{1}{4}\) can be rewritten to show a dependency on positive values of \x\ in a particular range.

Exponent Rules

Understanding exponent rules is crucial when tackling equations involving powers. Exponents represent how many times a number is multiplied by itself. Here are some key exponent rules that are often used:

• Negative Exponent Rule: \(x^{-n} = \frac{1}{x^n}\) transforms negative exponents into fractions, which simplifies the expression.
• Power of One: Any number raised to the power of one remains unchanged, i.e., \(x^1 = x\).
• Zero as Exponent: A number raised to the power of zero equals one, \(x^0 = 1\), except when \x\ is zero itself.

For the exercise \(x^{-3} = \frac{1}{4}\), applying the negative exponent rule allowed the equation to be rewritten as \(\frac{1}{x^3} = \frac{1}{4}\). This conversion simplifies the process of finding solutions further. By using exponent rules, equations become more manageable and allow for easier identification of potential solutions. Understanding these rules helps demystify complex algebraic expressions and aids in correctly manipulating equations to solve for unknowns.

Solving Equations

Solving equations involves finding the value(s) of variables that satisfy an equation. This process often requires a mix of arithmetic and algebraic manipulation.
To solve the equation \(x^{-3} = \frac{1}{4}\), follow these steps:

• Rewrite the equation: Transform it using exponent rules: \(\frac{1}{x^3} = \frac{1}{4}\).
• Cross-multiply: Simplify the equation to get rid of the fraction: \4 = x^3\.
• Isolate the variable: Solve for \x\ by taking the cube root of both sides: \x = \sqrt[3]{4}\.

The solution reveals that \(x = \sqrt[3]{4}\), which is greater than one. Therefore, the positive solution satisfies the condition \(a > 1\). Each stage in solving the equation is an application of algebraic techniques and logic. It is akin to unraveling a puzzle where structured steps lead to the final answer. Understanding these approaches strengthens one's ability to tackle a variety of mathematical problems efficiently.