Problem 22 - Algebra 2 and Trigonometry

Question

In \(18-23,\) solve for the variable in each equation. Express the solution to the nearest hundredth. $$ 5+b^{5}=56 $$

Step-by-Step Solution

Answer

\(b \approx 2.58\)

Step 1: Isolate the Exponential Term

The equation is given as \(5 + b^5 = 56\). Begin by isolating the exponential term \(b^5\) by subtracting 5 from both sides of the equation.\[ b^5 = 56 - 5 \]\[ b^5 = 51 \]

Step 2: Solve for the Variable

To solve for \(b\), take the fifth root of both sides of the equation to remove the exponent. \[ b = 51^{1/5} \]

Step 3: Calculate the Fifth Root

Use a calculator to approximate the value of \(51^{1/5}\). The calculation yields:\[ b \approx 2.58 \]

Key Concepts

Exponential Terms

In math, exponential terms refer to expressions where a number, known as the base, is raised to a power or an exponent. For example, in the equation given in the original exercise, the term \( b^5 \) involves an exponential term where \( b \) is the base and 5 is the exponent. This tells you that \( b \) is multiplied by itself five times. Understanding exponential terms is crucial because they appear frequently in mathematical equations and real-world applications, such as calculating compound interest or population growth.When handling equations with exponential terms, you usually need to simplify or manipulate the equation to isolate or solve for the variable. Exponents can turn simple arithmetic into more complex problems that require careful calculation. By isolating the exponential term, you create a simpler equation that can be more easily solved, as shown in the original exercise. Remember that the process of isolating any term in an equation involves performing inverse operations to simplify the equation.

Fifth Root Calculation

Calculating roots, like the fifth root, involves determining a number that, when raised to a specified power, results in the original number. In simpler terms, the fifth root of a number is a value that, when multiplied by itself five times, yields the original number.In our exercise, we had to find the fifth root of 51, noted as \( 51^{1/5} \). Here's a step-by-step on how to approach this:

• Use a calculator or a computer to find the approximate value, since manual calculation can be quite complex. Calculators often have a root function to make this easier.
• If you place 51 and input it into a calculator as a fifth root, you will find that the approximate value of \( 51^{1/5} \) is 2.58.
• Round the result to your desired decimal place, as instructed in the exercise (in this case, the nearest hundredth).

Calculating roots can be straightforward with proper tools, and the key here is to ensure precision by using a calculator for such calculations.

Isolating Variables

Isolating variables is a fundamental skill in algebra, where the goal is to "solve for" the variable by getting it alone on one side of the equation. This procedure often involves performing reverse operations to both sides of the equation in a balanced manner so that the equation stays true.Let's look at how you might do this with the equation \(5 + b^5 = 56\):

• First, identify the term involving the variable (in this case, \(b^5\)).
• To isolate \(b^5\), you subtract 5 from both sides: \[ b^5 = 56 - 5 \]
• This leaves you with \( b^5 = 51 \), effectively isolating the exponential term.
• From this point, you can then perform further steps to solve for \( b \), such as taking the fifth root as described in the previous section.

By isolating variables, not only do you clarify the equation, but you also make it solvable. Learning how to isolate variables is foundational and will make solving various algebraic problems much more manageable.