Problem 33 - Algebra and Trigonometry

Question

Solve each equation with rational exponents in Exercises \(31-40\) Check all proposed solutions. $$(x-4)^{\frac{3}{2}}=27$$

Step-by-Step Solution

Answer

The solution to the equation \((x-4)^{\frac{3}{2}} = 27\) is \(x = 13\).

Step 1: Isolate the term with the rational exponent

Start by isolating \( (x-4)^{\frac{3}{2}} \). In this case, it is already isolated.

Step 2: Apply the reciprocal of the exponent

The reciprocal of \(\frac{3}{2}\) is \(\frac{2}{3}\). Apply this power to both sides of the equation to cancel out the exponent on the left hand side. You'll get: \( ((x - 4)^{\frac{3}{2}})^{\frac{2}{3}} = 27^{\frac{2}{3}} \). This simplifies to \( x - 4 = \sqrt[3]{27^2} \).

Step 3: Simplify exponent on the right side

Now simplify the cubed root of \(27^2\) to get \(x - 4 = 3^2\), which further simplifies to \(x - 4 = 9\).

Step 4: Solve for x

Finally, solve for \(x\) by adding 4 to both sides of the equation: \(x = 13\).

Step 5: Check the solution

Substitute \(x = 13\) back into the original equation \( (x-4)^{\frac{3}{2}} = 27 \) and check if the equation holds true. If yes, then the solution is valid.

Key Concepts

Equation Solving

When solving equations involving rational exponents, one of the initial steps is to isolate the term that contains the exponent. In our exercise, the equation is already configured in a form where the exponent term,

• \((x-4)^{\frac{3}{2}}=27\)

was solitary on one side. This set-up simplifies your task as the only step you have to undertake next is addressing the rational exponent to simplify the equation further. Breaking down an equation to expose its core elements is a key phase in equation solving. Each variable is managed separately by performing inverse operations, and it becomes more straightforward to solve as each step progresses. Remember, patience and thoroughness are your best tools.

Reciprocal Exponents

A reciprocal exponent is basically the complementary fraction of an exponent. If you have an exponent such as

• \(a^{\frac{m}{n}}\),

its reciprocal would be

• \(a^{\frac{n}{m}}\).

Using reciprocal exponents aids in simplifying equations by essentially cancelling out the original exponent. In our example, the term within the equation

• \((x-4)^{\frac{3}{2}}\)

was raised by its reciprocal exponent,

• \(\frac{2}{3}\),

transforming the equation into a much simpler expression

• \(x-4\).

This approach not only simplifies calculations but also ensures the correctness of each step, leading to the final solution.

Exponent Laws

Exponent laws are crucial in manipulating and simplifying expressions involving powers. When dealing with expressions such as our given equation, several exponent laws come in handy:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \cdot n}\)
• Power of a Product: \((ab)^m = a^m \cdot b^m\)

In the problem, the power of a power law was pivotal when raising

• \((x-4)^{\frac{3}{2}}\)

to

• \(\frac{2}{3}\),

allowing for simplification into

• \(x-4\).

Understanding and applying these laws allows one to transform complex equations into something more manageable, guiding you step-by-step to the solution.

Substitution Method

The substitution method often comes in handy for verifying potential solutions to equations. Once a solution is derived, substituting it back into the original equation checks whether it satisfies all the conditions laid out by the equation.
In this instance, once we found

• \(x = 13\),

substituting it back into the original equation,

• \((x-4)^{\frac{3}{2}}=27\)

confirmed that

• the left side \((13-4)^{\frac{3}{2}}\)

equals

• 27,

proving our solution is correct. Substitution not only reassures the correctness of your derived solution but bolsters your understanding of each step worked through. It highlights any potential errors one might overlook during calculation, making it an invaluable step in equation solving.