Problem 70 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$(x+200)^{2 / 3}=36$$

Step-by-Step Solution

Answer

x = 16

Step 1: Isolate the exponent term

The given equation is \( (x+200)^{2/3} = 36 \). To isolate the term with the exponent, ensure it is on one side of the equation: \( (x+200)^{2/3} = 36 \).

Step 2: Eliminate the fractional exponent

To remove the fractional exponent, raise both sides of the equation to the power of \(3/2\): \[ \left((x+200)^{2/3}\right)^{3/2} = 36^{3/2} \]. This simplifies to \[ x+200 = 36^{3/2} \].

Step 3: Calculate the exponent

Calculate \( 36^{3/2} \). First, find the square root of 36, which is 6, and then raise 6 to the power of 3: \ 36^{3/2} = 6^3 = 216 \.

Step 4: Solve for x

With \( x + 200 = 216 \), isolate \( x \) by subtracting 200 from both sides: \[ x = 216 - 200 \]. Therefore, \ x = 16 \.

Key Concepts

Fractional Exponents

Understanding fractional exponents is essential for solving many algebraic equations. A fractional exponent, like \(\frac{2}{3}\), represents a power and a root. In this exercise, \((x+200)^{2/3}\), the exponent \(\frac{2}{3}\) means you first square the term \((x+200)\) and then take the cube root of the result. Breaking down the fractional exponent step-by-step helps demystify its meaning and application. Practicing with these types of exponents can significantly boost your precalculus skills.

Equation Isolation

Isolating the exponent term is one of the first steps in solving \( (x+200)^{2/3} = 36\). Start by ensuring the term involving the variable is by itself on one side of the equation. This makes it easier to handle subsequent operations. In our example, the equation is already set up nicely with \( (x+200)^{2/3} \) isolated. Once isolated, you can focus on removing the exponent and simplifying the equation. Always aim for a clear, isolated term to simplify further calculations.

Algebraic Simplification

After isolating the term, the next step involves removing the fractional exponent. You do this by raising both sides of the equation to the reciprocal power. In this case, raising both sides to the power of \( \frac{3}{2} \) simplifies \[ \left((x+200)^{2/3}\right)^{3/2} = 36^{3/2} \]. This eliminates the fractional exponent, transforming the equation to a simpler form: \ x + 200 = 216 \. Algebraic simplification is a key skill, allowing us to turn complex expressions into manageable ones.

Precalculus

This problem is a good example of precalculus concepts, combining several skills. It involves understanding and manipulating fractional exponents, isolating variables, and simplifying expressions. Mastering these steps equips you to handle more complex problems in calculus. Start by practicing simpler problems, and gradually move to more challenging ones. Precalculus lays the groundwork for your study of calculus, so it is important to grasp these fundamental concepts thoroughly.