Problem 21 - Algebra Form and Function

Question

In Exercises \(1-21,\) solve the equation for the variable. $$ 12=\sqrt{\frac{z}{5 \pi}} $$

Step-by-Step Solution

Answer

Answer: z = 720π

Step 1: Square both sides of the equation

In order to eliminate the square root on the right side of the equation, we will square both sides. This gives us: $$ (12)^2 = \left(\sqrt{\frac{z}{5 \pi}}\right)^2 $$

Step 2: Simplify the equation

After squaring both sides, we now have: $$ 144=\frac{z}{5 \pi} $$

Step 3: Multiply both sides by 5π

To eliminate the denominator in the fraction, we will multiply both sides by 5π: $$ 144(5 \pi) = \frac{z}{5 \pi}(5 \pi) $$

Step 4: Simplify and solve for z

The equation now becomes: $$ 720 \pi = z $$ So, the solution to the given equation is: $$ z = 720 \pi $$

Key Concepts

Squaring Both Sides

Sometimes, equations include a square root, which can make things tricky. To get rid of the square root, you can "square" both sides of the equation. This means multiplying each side by itself. Consider the equation \(12=\sqrt{\frac{z}{5 \pi}}\). Here, the right side has a square root.Squaring both sides looks like this:

• The left side: \((12)^2\).
• The right side: \(\left(\sqrt{\frac{z}{5 \pi}}\right)^2\).

This operation will cancel the square root on the right side, so it simplifies to the number or expression inside the root. After squaring, you will have \(144 = \frac{z}{5 \pi}\). This process helps to simplify the equation so you can eventually find the value of \(z\). Squaring both sides is a handy tool when you need to deal with square roots in equations.

Eliminating Square Roots

Eliminating square roots in equations is important because it allows you to turn complex expressions into simpler ones. When you have a square root like \(\sqrt{\frac{z}{5 \pi}}\), it might be a bit confusing to work with.To "eliminate" it, you perform an operation that makes the square root disappear—squaring the expression is the trick here. Squaring means multiplying the square root by itself, which simply gives you what was initially inside the square root. Once the square roots are gone, as in our example, you transition from \(12 = \sqrt{\frac{z}{5 \pi}}\) to the simplified form \(144 = \frac{z}{5 \pi}\) after squaring both sides. Now with no square roots left, solving the resulting equation becomes straightforward.

Multiplying to Clear Fractions

Fractions can be eliminated from an equation by multiplying both sides by the denominator of the fraction. This clears the fraction and simplifies the equation, making it easier to solve. For the equation \(144 = \frac{z}{5 \pi}\), the fraction is \(\frac{z}{5 \pi}\).By multiplying every term by \(5 \pi\), you effectively remove the fraction because:

• You multiply the left side \(144 \times 5 \pi\).
• The right side becomes \(\frac{z}{5 \pi} \times 5 \pi\), which simplifies to \(z\).

This math operation helps you isolate the variable \(z\) and ultimately find its value. Understanding how to clear fractions by multiplying both sides is practical for simplifying and solving equations that involve fractions in variable expressions.