Problem 30 - College Algebra

Question

Solve and check each equation with rational exponents. $$ x^{3 / 2}=27 $$

Step-by-Step Solution

Answer

The solution of the equation \( x^{3 / 2} = 27 \) is \( x = 9 \).

Step 1: Isolate the exponential term

The first step is to isolate the term with the exponential, which in this case is already done, as the equation is \( x^{3 / 2} = 27 \).

Step 2: Apply the square to both sides

In order to remove the exponent of \( \frac{3}{2} \), square both sides of the equation. Squaring the left side gives us \( (x^{3 / 2})^{2/3} = x \), and squaring the right side gives us \( 27^{2/3} \). This results in the new equation: \( x = 27^{2/3} \).

Step 3: Evaluate the right side

Calculate the value of \( 27^{2/3} \). The value of \( 27^{2/3} \) is 9. Therefore, the solution for x is 9.

Step 4: Substitute and verify

The last step is to substitute the value of x into the initial equation and verify that both sides are equal. Substituting x = 9 in the equation \( x^{3 / 2} = 27 \) gives \( 9^{3 / 2} = 27 \). After calculating, we see that indeed \( 27 = 27 \), so our solution is correct.

Key Concepts

Solving Equations

Solving equations involving rational exponents can initially seem daunting, but understanding the steps makes the process straightforward. Let's begin with a clear equation: \(x^{3/2} = 27\). The key to solving this equation is to "undo" the operation that is acting on \(x\). In this instance, the exponent \(3/2\) is our focus. To isolate \(x\), use the method of exponents. Specifically, you want to apply the inverse operation associated with a rational exponent. Since \(3/2\) is our exponent, take the reciprocal of it, which is \(2/3\). When you apply \((3/2) * (2/3)\) to \(x^{3/2}\), you simplify it to \(x\). Therefore, by raising both sides of the equation to \(2/3\), we clear the exponent on \(x\):

• Result on the left: \((x^{3/2})^{2/3} = x\)
• Apply the operation on the right: \(27^{2/3}\)

Ultimately, solving these equations requires analyzing the given operations and applying appropriate inverse methods.

Exponents

Exponents, particularly rational exponents, are crucial tools in algebra. A rational exponent, such as \(3/2\), can be expressed as a combination of roots and powers. This means \(x^{3/2}\) represents the square root followed by cubing or vice versa.Let's dissect \(x^{3/2}\). This notation implies two operations:

• First, \(x^{1/2}\) means "take the square root of \(x\)".
• Then, you apply another operation: raising the result of the root to the power of \(3\)".

This dual operation can be switched in sequence thanks to the associative properties of multiplication; whether you compute \((x^{3})^{1/2}\) or \((x^{1/2})^{3}\), you ultimately end up with the same result.Understanding rational exponents simplifies solving complex equations, as it allows us to convert roots into exponential forms, creating consistency across different problems.

Verification of Solution

Verification is a vital step after solving any equation. Checking ensures the obtained solution satisfies the original equation, affirming correctness. After solving \(x^{3/2} =27\) to find \(x = 9\), it's crucial to verify by substituting back into the initial equation: \(9^{3/2} = 27\).Breaking this down, execute:

• Calculate \(9^{1/2} \), which is the square root of 9, resulting in 3.
• Next, compute \(3^3\), which equals 27.

Both sides of the equation should match, confirming the solution is correct. This process not only validates the mathematical result but also builds confidence in algebraic techniques. Always make this a concluding step in solving equations.