Problem 70 - Algebra 1

Question

The geometric mean of 16 and a is 32. What is the value of a?

Step-by-Step Solution

Answer

The value of a is 64.

Step 1: Write down the mathematical equation

The given problem says that the geometric mean of 16 and a is 32, which we can write down as \( \sqrt{16a} = 32 \).

Step 2: Square both sides of the equation to eliminate the square root

We square both sides of the equation in order to eliminate the square root. This gives us \( ( \sqrt{16a} )^{2} = 32^{2} \), which simplifies to \( 16a = 1024 \).

Step 3: Isolate the variable a

To solve for a, we need to isolate it. We divide both sides of the equation by 16. This gets us the result: a = \( 1024 / 16 = 64 \).

Key Concepts

Square Root

Understanding the concept of a square root is fundamental when handling problems involving the geometric mean. The square root function is the inverse of squaring a number. For example, if you square 5, you get 25, and hence, the square root of 25 is 5. This is written as \( \sqrt{25} = 5 \). In our exercise, where the geometric mean of 16 and a variable 'a' is given as 32, we represent it using a square root, because the geometric mean of two numbers is defined as the square root of their product.

In the given solution, the first step is to express this definition algebraically as \( \sqrt{16a} = 32 \). The purpose of squaring both sides in step 2 isn't arbitrary. It's a deliberate move to eliminate the square root, making the equation simpler to solve. This is a common strategy when dealing with equations involving square roots and is often the first step in solving such problems.

Isolating Variables

After clearing the square root by squaring both sides, we often must isolate the variable to find its value. Isolating the variable means rearranging the equation so that the variable we're solving for is by itself on one side of the equation. In the context of our exercise, after squaring both sides, our equation becomes \( 16a = 1024 \).

To isolate 'a', we need to get rid of everything that's attached to it. Here, 'a' is being multiplied by 16, so the opposite operation, division, will be used to isolate 'a'. Dividing both sides by 16 gives us the value of 'a' alone on one side of the equation: \( a = \frac{1024}{16} \). This is the third step in our step-by-step solution, leading straight to the answer. Mastering the ability to isolate variables is crucial for solving algebraic equations of all types, not just in the context of geometric means.

Solving Algebraic Equations

Solving algebraic equations is a sequence of steps to find the value(s) of the variable(s) that satisfy the given equation. The geometric mean problem we've discussed is just one instance of an algebraic equation. The general strategy is first to simplify the equation, if necessary, then use mathematical operations to isolate the variable.

After simplifying and isolating 'a' in our example, we perform the division and conclude that \( a = 64 \). This process of first eliminating the square root by squaring, then isolating the variable, follows the standard method of solving algebraic equations. It's about maintaining balance — what you do to one side of the equation; you must do to the other to keep the equation true. Understanding and applying these steps ensures that you can solve algebraic equations confidently and correctly.