Problem 32 - Algebra and Trigonometry

Question

Solve each equation with rational exponents in Exercises \(31-40\) Check all proposed solutions. $$x^{\frac{3}{2}}=27$$

Step-by-Step Solution

Answer

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

Step 1: Understand the equation

The given equation is \(x^{\frac{3}{2}}=27\). This equation says that if you apply an operation (taking the square root followed by a cube) to some number \(x\), you get 27. You need to find the \(x\) that makes this true.

Step 2: Get rid of the exponent

To eliminate the fraction exponent, raise both sides of the equation to the reciprocal of the exponent (in this case \(\frac{2}{3}\)). The equation becomes \((x^{\frac{3}{2}})^{\frac{2}{3}} = 27^{\frac{2}{3}}\). By the power rule, the left side simplifies to \(x\). The right side simplifies to 9 using the cube root and then squaring.

Step 3: Confirm the solution

It is important to substitute the proposed solution back into the original equation to make sure it actually works. Substitute \(x = 9\) into the original equation. The left hand side becomes 9 to the power of \(\frac{3}{2}\), or 27, which matches the right-hand side, confirming that \(x = 9\) is indeed the solution.

Key Concepts

Rational Exponents

When encountering an equation with rational exponents, such as \(x^{\frac{3}{2}} = 27\), it's imperative to understand rational exponents fundamentally. A rational exponent is an exponent that is a ratio of two integers, where the numerator indicates the power and the denominator the root.

For example, \(x^{\frac{3}{2}}\) means that \(x\) is firstly squared and then the cube root is taken. To solve for \(x\), you must 'undo' this operation. In practice, you apply the reciprocal of \(\frac{3}{2}\), which is \(\frac{2}{3}\), effectively reversing the operation and thus isolating \(x\). This can sometimes be perceived as a 'double operation'—combining both square and cube root in a specific sequence, which is pivotal for grasping the subsequent steps.

Power Rule

The power rule is a mathematical theorem utilized to simplify expressions with exponents. It dictates that when a term with an exponent is raised to another exponent, you multiply the exponents. Using the equation \(x^{\frac{3}{2}} = 27\) as an example, the power rule is applied when you raise both sides to the exponent \(\frac{2}{3}\).

Applying the power rule gives us \((x^{\frac{3}{2}})^{\frac{2}{3}} = x^{(\frac{3}{2}\cdot\frac{2}{3})} = x^1 = x\), successfully isolating the variable. It's essential to make these steps intuitively understood, and recognizing patterns can streamline solving similar equations in the future. In the equation's context, the power rule gives us the key to unlock the value of \(x\) from its exponentiated form.

Cube Root

The cube root function is the inverse operation of raising a number to the third power. For instance, the cube root of 27 is 3, since \(3^3 = 27\). In our equation, after applying the reciprocal exponent of \(\frac{2}{3}\) to 27, we first tackle the cube root of 27, which simplifies to 3.

Understanding cube roots is fundamental when dealing with rational exponents that imply root operations. It's through this knowledge that you can simplify the right side of the equation to 9 in our example, by squaring the cube root of 27, which is 3. The key takeaway is the sequential nature of operations when dealing with rational exponents; the cube root is taken first, followed by squaring to attain the simplified result.