Exponential Equations - College Algebra

TL;DR

Learn how to solve exponential equations by using cross multiplication, combining fractions, factoring, and substitution.

Transcript

in this video we're going to talk about how to solve this particular exponential equation which can be quite challenging but for those of you who want to try this problem feel free to pause the video and work on it so let's begin since we have two fractions separated by an equal sign i recommend that we begin by cross multiplying the two fractions ... Read More

Key Insights

• 😵 Cross multiplying fractions and simplifying is an essential step in solving exponential equations with fractions.
• 😑 Using the rules of exponents, you can replace equivalent exponential expressions to simplify the equation further.
• 🙃 Multiplying both sides by a particular term can help eliminate fractions in the equation.
• ❓ Factoring is an effective strategy to solve trinomials in exponential equations.
• 👻 Substitution with a new variable allows for easier factoring and solution finding.
• 0️⃣ The zero product property is applied to set each factor in the equation equal to zero, facilitating solution finding.
• ✅ Checking the solution by substituting it back into the original equation ensures its accuracy.

Questions & Answers

Q: How do you begin solving an exponential equation with fractions?

To start, cross multiply the fractions and simplify. For example, 10(4^x - 4^(-x)) = 5(3) becomes 10(4^x)/10 = 3/2.

Q: Can you replace equivalent exponential expressions in the equation?

Yes, you can replace 4^x with (4^2x)/(4^x) and 4^(-x) with 1/(4^x) based on the rules of exponents.

Q: What is the next step after adjusting the equation?

Multiply both sides by 4^x to eliminate the fraction, resulting in 2(4^(2x)) - 3(4^x) - 2 = 0.

Q: How do you factor the resulting expression?

Multiply the leading coefficient (-2) by the constant term (-2) to get +4. Find two numbers that multiply to 4 and add to the middle term (-3). In this case, -4 and 1 work.

Q: How can you check the solution to the exponential equation?

Substitute the found value back into the original equation and simplify both sides. If they are equal, the solution is correct.

Summary & Key Takeaways

• To solve the exponential equation with fractions, start by cross multiplying the fractions and simplifying.

• Adjust the equation by replacing equivalent exponential expressions and applying the rules of exponents.

• Multiply both sides of the equation to eliminate the fraction and simplify.

• Factor the resulting expression and make a substitution to further simplify the equation.

• Solve for the variable by setting each factor equal to zero using the zero product property.

• Check the solution by substituting it back into the original equation and ensuring both sides are equal.