Problem 61 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$\sqrt[4]{x-15}=2$$

Step-by-Step Solution

Answer

x = 31

Step 1 - Isolate the Radicand

The equation given is \(\frac{x-15}{4} = 2\). In this step, notice that \(\frac{(.)}{4}\) represents a fourth root. Therefore, isolate the radicand by raising both sides of the equation to the power of 4 to eliminate the fourth root: \[ \left(\frac{x-15}{4}\right)^4 = 2^4 \]

Step 2 - Simplify Both Sides

Simplify both sides of the equation from Step 1. Simplifying the right side: \(\frac{2}{4} = 16\), so \[x - 15 = 16 \]

Step 3 - Solve for x

To isolate \(x\), add 15 to both sides of the equation from Step 2: \[x - 15 + 15 = 16 + 15\] Simplify to isolate \(x\): \[ x = 31\]

Key Concepts

Isolating the Radicand

To start solving radical equations, the very first step involves isolating the radicand. The radicand is the expression inside the root symbol. For the equation \(\frac{x-15}{4}=2\), you should recognize that the symbol stands for 'fourth root'. In this specific exercise, our radicand is \(x-15\). By isolating the radicand, we can more easily manipulate the equation. To remove the fourth root, we need to neutralize it by performing an exponentiation.

Exponentiation

Exponentiation is the mathematical operation used to manage powers and roots. To isolate the radicand in our exercise equation, we raise both sides of the equation to the power of 4. This step eliminates the fourth root. Here’s how it looks: \[ \left\(x-15\/4\right\)^4 = 2^4\] Exponentiation ensures that \(x-15\) is no longer under the root, simplifying our equation. Now, perform the power on the right side: \[2^4 = 16\] This changes the equation to: \[x-15=16\]

Simplifying Equations

Once exponentiation is complete, the next step is simplifying both sides of the equation. In this example, after exponentiating, we have \((x-15\quad=\quad 16)\). Simplifying the right side of the equation to 16 makes it easier to solve for the unknown variable. The process of simplifying helps us break down complex terms into simpler, more manageable ones.

Fourth Root

Understanding the fourth root is crucial when solving equations involving roots. The fourth root of a number \(a\) is a value \(b\) such that \(b^4 = a\). For instance, in the equation \(x-15=16\), the fourth root \(2\), when raised to the power of 4, returns 16, as shown: \[2^4 = 16\] Accurately calculating and using fourth roots helps in isolating and solving for the unknown variable efficiently.