Algebra: Equations with Exponents and Simplifying Rational Equations

Q: What is the best way to solve a quadratic equation?

Use factoring, completing the square, or the quadratic formula. The method depends on the specific form of the quadratic equation.

ElevatEd
Math
August 24, 2024

Interesting fact about algebra is it has both variables and numerals in simple words it is a combination of alphabets and numbers, and these are employed in mathematic equations.

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Algebra is a branch of mathematics that deals with symbols and rules for manipulating those symbols. Learn more about how algebra allows us to solve equations and understand relationships between variables, laying the foundation for more advanced mathematical concepts and real-world problem-solving with 98thPercentile.

Equations and Exponents

Variables raised to power are found in equations involving exponents. To assist you in solving these kinds of equations, consider the following procedures and examples:

Remove the Exponential Term: If any terms in the equation don't have an exponent, shift them to the opposite side.
Rewrite the equation: To handle the exponent use logarithms if needed.
Solve for the Variable: Once the exponential term is isolated, solve for the variable.

Example 1: Basic Exponential Equation

If base numbers are equal, then their powers must be equal

2𝑥 = 16

2𝑥 = 24

𝑋 = 4

Example 2:

42𝑥+1 = 64

42𝑥+1 = 43

2𝑥+1 = 3

2𝑥 = 3−1

2𝑥 = 2

𝑥= 22=1

∴ 𝑥 = 1

Simplifying Rational Equations

Rational equations involve fractions with polynomials in the numerator and denominator. Here are steps to simplify and solve them:

Find a Common Denominator: To combine fractions, find the least common denominator (LCD).
Clear the Fractions: Multiply both sides of the equation by the LCD to eliminate the denominators.
Simplify and Solve: Simplify the resulting equation and solve for the variable.

Example: Simplifying a Rational Equation

Let's take equation 𝑥𝑥+2+3𝑥−2=2𝑥+5𝑥2−4

Factor the denominator on the right side 𝑥2−4=(𝑥+2) (𝑥−2)
The common denominator is (𝑥+2) (𝑥−2)
Multiply every term by the common denominator

𝑥(𝑥−2)+ 3(𝑥+2)= (2𝑥+5)

= 𝑥2−2𝑥+3𝑥+6 = 2𝑥+5

=𝑥2+𝑥+6 = 2𝑥+5

Move all terms to one side of the equation

𝑥2−𝑥+1 = 0

Solve the quadratic equation using the quadratic formula

𝑥=−𝑏±√𝑏2−4𝑎𝑐2𝑎

For a=1, b=−1, and c=1

𝑥=−1±√1−42

𝑥=−1±√32

𝑥=1±𝑖√32

∴ 𝑥=1+𝑖√32 (𝑜𝑟) 𝑥=1−𝑖√32

Simplifying rational equations and understanding equations with exponents will improve mathematical understanding along with enhanced problem-solving skills.

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FAQ’s (Frequently Asked Questions)

Q1: How do I handle exponents with different bases?

Ans: There are two ways to do this. Either rewrite the numbers with the same base or use logarithms to solve the equation.

Q2: What if the equation involves negative exponents?

Ans: Negative exponents can be converted to positive by taking the reciprocal.

Q3: How do I simplify a complex rational equation?

Ans: Factor all polynomials in the numerator and denominator, find the LCD, and multiply through to clear the fractions. Simplify the resulting equation.

Q4: What is the best way to solve a quadratic equation?

Ans: Use factoring to complete the square or the quadratic formula. The method depends on the specific form of the quadratic equation.

Q5: Can exponential equations have no solutions?

Ans: Yes, exponential equations can have no solutions, especially if the base is positive and the equation is set equal to a negative number.