Problem 12 - Intermediate Algebra

Question

Fill in the blanks. Use a calculator to determine whether 2.5646 is an approximate solution of \(2^{2 x+1}=70\)

Step-by-Step Solution

Answer

2.5646 is approximately a solution because \(2^{6.1292}\) is close to 70.

Step 1: Substitute the Value

First, we need to substitute the given approximate solution value, 2.5646, into the equation \(2^{2x+1}\). Set \(x = 2.5646\) and compute the expression.

Step 2: Calculate the Exponent

Substitute \(x = 2.5646\) into the exponent: \(2(2.5646) + 1\). This calculation results in \(5.1292 + 1 = 6.1292\). This number is the power to which 2 will be raised.

Step 3: Evaluate the Power of 2

Calculate \(2^{6.1292}\) using a calculator. This involves raising 2 to the power computed in the previous step.

Step 4: Compare with 70

Check whether the calculated value approximately equals 70. The aim is to see how close \(2^{6.1292}\) is to 70.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers, called the base and the exponent. The base is the number that is multiplied, while the exponent indicates how many times the base is used in the multiplication. For an expression like \(2^x\), 2 is the base and \(x\) is the exponent. It represents \(2 \times 2 \times ...\) (repeated \(x\) times).

• Example: \(2^3 = 2 \times 2 \times 2 = 8\).
• Special Cases: \(2^0 = 1\) since any number to the power of zero is 1.
• Fractional exponents, like in \(2^{1/2}\), represent roots, such as square roots.

When working with non-integer exponents, like \(2^{6.1292}\) in the problem, it is essential to understand that it involves fractional multiplication. Calculators are typically used in these situations.

Substitution Method

The substitution method in algebra involves replacing variables with their values. It helps to simplify equations by substituting a number for a variable. Let's see how it's done.
When using the substitution method:

• Identify the variable in the equation.
• Replace the variable with the given value.
• Simplify the equation to solve it.

In the original problem, the substitution method is used by taking\(x = 2.5646\) and placing it into the equation \(2^{2x+1}\). This transforms the expression into a numerical form that can then be calculated using a calculator or manually, if feasible.
Always double-check your substitution and subsequent calculations to ensure accuracy.

Calculator Use in Mathematics

Calculators are vital tools in mathematics, especially when dealing with complex calculations or non-integer numbers as we encounter in this problem. Using a calculator helps to handle large numbers and complicated functions efficiently.
Here’s how they are typically used:

• Scientific calculators can handle exponentiation, trigonometric functions, and logarithmic calculations with ease.
• Enter numbers carefully and check your calculator's settings for necessary configurations (e.g., degrees vs. radians for angle measure).
• Understand the buttons: exponentiation is usually \(y^x\) or a similar key, depending on the calculator.

For an exercise like evaluating \(2^{6.1292}\), a calculator minimizes the risk of human error, providing rapid and accurate results. Always re-calculate if the results seem off to confirm your findings.