91Ó°ÊÓ

When we talk about solving equations, we're finding the values of the variable that make the equation true. In our problem, the goal is to find the value of x that satisfies the original equation. Here are the steps involved:

• Identify what needs to be solved
• Manipulate the equation to isolate the variable
• Verify your solutions by substituting back into the original equation

In our specific problem, the equation: \( (x^{2} - 16)^{1 / 2} = 9 \) involves a square root. We begin by squaring both sides to eliminate the square root. Always treat both sides of the equation equally, maintaining balance. For this problem, it’s \( (\text{x}^{2} - 16) = 9^{2} \).
Next, simplify it to \( x^{2} - 16 = 81 \) and solve for the variable, x. From here, add 16 to both sides to further isolate x, providing us \( x^{2} = 97 \). By taking the square root of both sides, we obtain \( x = \pm \sqrt{97} \). Solving equations is just about logically manipulating the existing equation in a series of orderly steps.

Square roots are the numbers that, when multiplied by themselves, give the original number. For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9. In mathematical notation, this is written as \( \sqrt{9} = 3 \).
In our equation, we dealt with a square root at the very start: \( (x^{2} - 16)^{1 / 2} = 9 \). Here’s what’s happening step-by-step:

• The expression inside the square root (\text{x}^{2} - 16) should be isolated first
• Square both sides to cancel the square root. Remember, squaring means raising to the power of 2, which 'undoes' the square root
• This results in a simpler equation without square roots and makes further operations easier

By understanding square roots, you recognize how to properly address and eliminate them during equation solving.

An irrational number is a number that cannot be expressed as a simple fraction. Irrational numbers have non-repeating, non-terminating decimal parts.
In our solution, we obtained solutions in terms of \( \sqrt{97} \), which is an irrational number. It approximates to 9.8488578... (and continues indefinitely).
Here’s why it's crucial:

• Irrational numbers can’t be precisely written as a fraction of two integers
• We often encounter square roots of non-perfect squares when solving equations
• When simplifying solutions involving square roots, keep them in root form when possible for exact values

So, for \( x^{2} = 97 \, x \pm \sqrt{97} \). Recognizing and correctly handling irrational numbers ensures we interpret and report solutions accurately.