Problem 102 - Precalculus

Question

Solve the equation \(10^{x^{2}}=0.001^{x} .\) Using a graphing calculator, plot the graphs \(y=10^{x^{2}}\) and \(y=0.001^{x}\) in the same viewing rectangle. Does this confirm your solution?

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = -3\), confirmed by graph intersections.

Step 1: Convert 0.001 to Power of 10

First, note that the number 0.001 can be rewritten as a power of 10. Specifically, \(0.001 = 10^{-3}\). Substituting this into the equation gives us \(10^{x^2} = (10^{-3})^x\).

Step 2: Simplify the Equation

Rewrite the equation using the exponentiation rule \((a^m)^n = a^{mn}\). This yields \(10^{x^2} = 10^{-3x}\). Since the bases are the same, we can equate the exponents: \(x^2 = -3x\).

Step 3: Set Up a Quadratic Equation

Move all terms to one side of the equation to form a quadratic equation: \(x^2 + 3x = 0\).

Step 4: Factor the Quadratic Equation

Factor the quadratic equation: \(x(x + 3) = 0\). This gives us two possible solutions for \(x\): \(x = 0\) or \(x = -3\).

Step 5: Plot the Graphs

Using a graphing calculator, plot the graphs of \(y = 10^{x^2}\) and \(y = 0.001^x\).

Step 6: Compare Graph Intersections with Solutions

Observe the intersection points of the graphs. They should intersect at the solutions previously found: \(x = 0\) and \(x = -3\). This confirms that our algebraic solutions match the graphical intersections.

Key Concepts

Exponents

Understanding exponents is crucial when solving equations like the one in our original exercise. An exponent represents how many times a number, known as the base, is multiplied by itself. For instance, in the expression \(10^{x^2}\), "10" is the base, and "\(x^2\)" is the exponent. It tells us that 10 is multiplied by itself \(x^2\) times.

Here are a few helpful tips to remember about exponents:

• If the exponent is zero, any non-zero base raised to this power equals one (e.g., \(a^0 = 1\)).
• When multiplying with the same base, add the exponents (e.g., \(a^m \times a^n = a^{m+n}\)).
• For raising a power to another power, multiply the exponents (e.g., \((a^m)^n = a^{m \times n}\)).

In our specific problem, we used the exponent rule to simplify \(10^{x^2} = (10^{-3})^x\) to \(10^{x^2} = 10^{-3x}\), allowing us to easily compare and manipulate the exponents directly.

Graphing Calculator

A graphing calculator is a powerful tool to visualize complex mathematical equations. It is designed to plot graphs, solve equations, and perform various calculations. In mathematical problems, like the one we have, a graphing calculator helps verify our algebraic solutions by providing a visual representation.

When using a graphing calculator:

• Enter the equations accurately, as precision is key to obtaining correct intersections.
• Set the viewing window appropriately. This ensures you capture all relevant details and intersection points.
• Use the graph feature to plot each function on the same set of axes, which allows for clear observation of where the graphs intersect.

In our exercise, inputting \(y = 10^{x^2}\) and \(y = 0.001^x\), we can visualize their behavior. Seeing the points where these curves cross confirms the intersection at our calculated solutions, \(x = 0\) and \(x = -3\).

Intersections of Graphs

The intersection of graphs refers to the points where two graphs meet or cross each other. These points represent the solutions to systems of equations, which are sets of equations with multiple unknowns.

In our equation \(10^{x^2} = 0.001^x\), it can be reimagined as finding the intersection of the graphs \(y = 10^{x^2}\) and \(y = 0.001^x\). The intersection points are the values of \(x\) that satisfy both equations simultaneously.

To analyze graph intersections, keep in mind:

• Each intersection point corresponds to a common solution for the equations represented by the graphs.
• If graphs do not intersect, there is no common solution.
• Having an algebraic solution aids in setting expectations, allowing you to confirm solutions found graphically.

For our problem, identifying the intersection points \(x = 0\) and \(x = -3\) on the graphing calculator confirmed our algebraic solutions, providing a holistic understanding of the equation's solutions.