Problem 66 - College Algebra with Modeling and Visualization

Question

Solve the equation. Check your answers. $$ x^{2 / 3}=16 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 64\).

Step 1: Understanding the Problem

The goal is to solve for \(x\) in the equation \(x^{2/3} = 16\). This means we need to find the value(s) of \(x\) that satisfy the equation when raised to the power of \(2/3\) and equals 16.

Step 2: Isolate the Power

Since the left side of the equation is \(x^{2/3}\), we can solve for \(x\) by raising both sides of the equation to the reciprocal power, which is \(3/2\). This will help in cancelling out the power on \(x\).

Step 3: Apply Reciprocal Power

Raise both sides of the equation to the power of \(3/2\):\[(x^{2/3})^{3/2} = 16^{3/2}\]This simplifies to:\[x = (16^{3/2})\].

Step 4: Simplify the Right Side

We need to calculate \(16^{3/2}\). First, recall that \(16^{1/2} = \sqrt{16} = 4\). Now, raise this result to the power of 3:\[16^{3/2} = (16^{1/2})^3 = 4^3 = 64\].

Step 5: Verify the Solution

Now we need to verify our solution. Plug \(x = 64\) back into the original equation: \[(64)^{2/3}\]. First, find \(64^{1/3} = \sqrt[3]{64} = 4\). Then, \(4^2 = 16\), which matches the right side of the equation. Thus, \(x = 64\) is correct.

Key Concepts

Exponents

Exponents tell us how many times to multiply a number by itself. In algebra, they are crucial in simplifying expressions and solving equations. For instance, when we see an exponent like \( x^{2/3} \), it indicates two operations:

• The "2" represents squaring, or multiplying the base by itself once.
• The "3" indicates the cube root, which is finding a number that, when multiplied by itself twice, gives back the original number.

In the equation \( x^{2/3} = 16 \), the power \( 2/3 \) suggests taking \( x \), raising it to the 2nd power (square), and then finding the cube root of that result to equal 16. Understanding these steps is essential, as it helps to unravel how the solution develops. Breaking down the exponent fraction into these operations aids in visualizing the transformation required to isolate \( x \).
This approach is integral in understanding how to manipulate and solve equations involving fractional exponents efficiently.

Verify Solution

Verifying a solution means checking to ensure that the solution we have satisfies the original equation. It's like double-checking our work.
In this context, after determining that \( x = 64 \) satisfies the equation \( x^{2/3} = 16 \), we must substitute 64 back into the equation to see if it holds true.

• The power \( 2/3 \) means we first find \( 64^{1/3} \) or \( \sqrt[3]{64} \), which turns out to be 4 because \( 4 \times 4 \times 4 = 64 \).
• Then, we square this result, \( 4^2 = 16 \).

The result matches the right side of the equation, confirming that our solution is correct. Verification gives us confidence in our answer. It ensures we haven't made a calculation mistake along the way and confirms the logical consistency of our approach. This is a practice that can greatly enhance accuracy when solving algebraic equations.

Reciprocal Powers

Reciprocal powers are used to simplify and solve equations involving exponents. They are particularly helpful when the equation includes fractional exponents. The reciprocal of an exponent is simply "flipping" the fraction over.
For example, in the equation \( x^{2/3} = 16 \), we employed the reciprocal exponent \( 3/2 \).

• This means raising both sides of the equation such that the powers can cancel out.
• For \( x^{2/3} \), the reciprocal \( 3/2 \) cancels \( 2/3 \), returning us to \( x^1 \) or \( x \).

This method clears the way to find \( x \). Applying the reciprocal power to both sides results in \( x = (16^{3/2}) = 64 \), allowing us to identify the original value. Mastering reciprocal powers is essential, as they unlock more complex calculations, creating a path to the simplest form of the variable. It makes solving polynomial equations with fractional exponents a much more straightforward process.