Problem 34 - Algebra and Trigonometry

Question

Solve each equation with rational exponents in Exercises \(31-40\) Check all proposed solutions. $$(x+5)^{\frac{3}{2}}=8$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 3\)

Step 1: Isolate the term with the rational exponent

First, keep the term \( (x+5)^{\frac{3}{2}} \) separate: \n $$(x+5)^{\frac{3}{2}} = 8 $$

Step 2: Get rid of the exponent

To get rid of the exponent, raise both sides to the reciprocal of \(\frac{3}{2}\), that is, \(\frac{2}{3}\). This gives: \n $$ ((x+5)^{\frac{3}{2}})^{\frac{2}{3}} = 8^{\frac{2}{3}} $$ \n Which simplifies to: \n $$ x+5 = (8^{\frac{2}{3}}) $$

Step 3: Calculate \(8^{\frac{2}{3}} \)

To calculate \(8^{\frac{2}{3}}\), first calculate the cube root of 8, which is 2, and then square the result: \n $$ 8^{\frac{2}{3}} = 2^2 = 4 $$

Step 4: Isolate x

Solving the equation \(x + 5 = 4\), yields to: \n $$ x = 4 - 5 $$ \n Which simplifies to: \n $$ x = -1 $$

Step 5: Check the solution

Substitute \(x = -1\) in the original equation and see if both sides are equal: \n $$$$ ((-1 + 5)^{\frac{3}{2}}) = 8^{\frac{2}{3}} $$$$ \n The left side simplifies to \(16\), and the right side to \(4\). Since these do not match, \(x = -1\) is not a solution. We must reconsider Step 2: \n $$ ((x+5)^{\frac{3}{2}})^{\frac{2}{3}} = (8^{\frac{2}{3}}) ^{\frac{3}{2}} $$ \n This gives: \n $$ x+5 = 2^3 = 8 $$ \n Now when x is isolated just like in Step 4 it gives: \n$$ x = 8 - 5 = 3 $$ \n When \(x = 3\) is substituted into the original equation the equality holds, thus \(x = 3\) is a solution of the equation

Key Concepts

Solving Equations

To solve equations with rational exponents, it’s essential to start by isolating the term that includes the exponent. Once it's isolated, you need to remove the exponent by using the reciprocal of the given rational exponent.
This involves raising both sides of the equation to this reciprocal power, effectively canceling the original exponent. For example, given \((x+5)^{\frac{3}{2}} = 8\), you would raise both sides by \(\frac{2}{3}\).
The resulting equation is simpler and only involves taking roots and powers. This method ensures that we simplify the problem in two steps: neutralizing the exponent and then solving the remaining simpler equation.
Here are the key steps you should always take:

• Isolate the term with the rational exponent
• Raise both sides to the reciprocal of the exponent
• Solve the simplified equation by removing leftover constants

Rational equations can sometimes have solutions that don’t satisfy the original equation, so always check your solutions!

Powers and Roots

Understanding the relationship between powers and roots is crucial for working with rational exponents. A rational exponent, like \(\frac{3}{2}\), means you first take the power (numerator) and then the root (denominator), or vice versa.
For instance, in \( (x+5)^{\frac{3}{2}} \), the expression suggests taking the square root (since the denominator is 2) and then cubing the outcome (numerator is 3) or simply combining them in one operation. Step by Step:

• Identify the operation: numerator is power, denominator is root
• Apply them in a way that simplifies the calculations

As an example, when you calculate \(8^{\frac{2}{3}} \), you should first see it as taking the cube root of 8 which is 2, and then squaring 2 to find 4. Thus, it’s both a method for simplifying terms and computing values easily.

Exponent Rules

Exponent rules are helpful tools for simplifying algebraic expressions involving powers. Some of the essential rules include the product of powers, power of a power, and the quotient of powers. These rules also apply when dealing with rational exponents.
When combining exponents, use the basic rule that says \((a^m)^n = a^{m \times n}\). This is particularly useful when you have an expression like \(((x+5)^{\frac{3}{2}})^{\frac{2}{3}}\). Here you multiply the exponents \(\frac{3}{2} \times \frac{2}{3} = 1\), effectively eliminating the powers.
Involvement of rational exponents:

• Apply the power of a power rule for simplification
• Use reciprocal exponents carefully to neutralize terms

Additionally, remember that when converting a number like \(8\) raised to the power of any rational exponent, it’s possible to break it down into more manageable calculations using these rules. This makes seemingly complex problems much simpler.