Problem 25 - Algebra for College Students

Question

Solve each of the equations. $$ \left(4^{x}\right)\left(16^{\frac{3 x-1}{x}}\right)=8 \quad\left\\{\frac{1}{2}\right\\} $$

Step-by-Step Solution

Answer

Solutions: \(x = \frac{-7 \pm \sqrt{57}}{2}\)

Step 1: Express Bases with a Common Base

Rewrite the numbers 4, 16, and 8 using the base of 2: - 4 can be expressed as \(2^2\). - 16 can be expressed as \(2^4\). - 8 can be expressed as \(2^3\). This simplifies our equation to: \((2^2)^x \, (2^4)^{\frac{3x-1}{x}} = 2^3 \, \frac{1}{2}\)

Step 2: Simplify Exponents

Use the rule \((a^m)^n = a^{mn}\) to simplify the exponents:- \((2^2)^x\) becomes \(2^{2x}\). - \((2^4)^{\frac{3x-1}{x}}\) becomes \(2^{\frac{16x-4}{x}} = 2^{16 - \frac{4}{x}}\). Our equation becomes:\(2^{2x} \cdot 2^{16 - \frac{4}{x}} = 2^{3} \cdot 2^{-1}\)

Step 3: Combine the Exponents

Use the property \(a^m \cdot a^n = a^{m+n}\) to combine the exponents on the left side:\(2^{2x + 16 - \frac{4}{x}} = 2^{2}\) Here, we have combined the powers into one base of 2.

Step 4: Equate the Powers

Since the bases are the same, we can set their exponents equal to each other:\(2x + 16 - \frac{4}{x} = 2\)

Step 5: Clear the Fraction

Multiply through by \(x\) to eliminate the fraction:\(2x^2 + 16x - 4 = 2x\) Simplifying, we get:\(2x^2 + 14x - 4 = 0\)

Step 6: Solve the Quadratic Equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve for \(x\) where \(a = 2\), \(b = 14\), and \(c = -4\):\(x = \frac{-14 \pm \sqrt{14^2 - 4 \cdot 2 \cdot (-4)}}{2 \cdot 2}\) Simplify further:\(x = \frac{-14 \pm \sqrt{196 + 32}}{4}\)\(x = \frac{-14 \pm \sqrt{228}}{4}\) Since \(\sqrt{228} = \sqrt{4 \times 57} = 2\sqrt{57}\):\(x = \frac{-14 \pm 2\sqrt{57}}{4}\)\(x = \frac{-7 \pm \sqrt{57}}{2}\) Thus, the solutions are \(x = \frac{-7 + \sqrt{57}}{2}\) and \(x = \frac{-7 - \sqrt{57}}{2}\).

Key Concepts

Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. Quadratic equations can be solved by several methods: factoring, completing the square, and using the quadratic formula. In many scenarios, the quadratic formula is the most straightforward approach. The quadratic formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] This formula allows you to find the exact solutions or roots of the quadratic equation by substituting the values of \(a\), \(b\), and \(c\). Remember:

• \(b^2 - 4ac\) is known as the discriminant. It determines the nature of the roots. - If it’s positive, you'll have two real and different roots. - If it's zero, you'll have one real root (a repeated root). - If it's negative, the roots are complex or imaginary.
• The solutions can also be fractions or involve square roots, especially when the discriminant isn't a perfect square.

This formula is a powerful tool in solving quadratic equations, ensuring accuracy and completeness of the solution.

Simplifying Exponents

Simplifying exponents involves using rules and properties of exponents to rewrite expressions in a simpler form. Exponents indicate how many times a number, known as the base, is multiplied by itself. Let’s explore some fundamental rules:

• \( (a^m)^n = a^{m \times n} \) is known as the power of a power rule. It shows that you multiply the exponents when raising a power to another power.
• \(a^m \times a^n = a^{m+n} \) is the product of powers rule. This allows you to add the exponents if you are multiplying similar bases.
• \(a^m / a^n = a^{m-n} \) is the quotient of powers rule. Subtract exponents when dividing powers with the same base.
• \(a^0 = 1\) for any non-zero base \(a\), indicating that any base raised to the zero power is 1.

By applying these rules effectively, you can transform complex expressions into more manageable forms. Simplifying exponents is foundational for solving algebraic problems involving exponentials.

Common Bases

Common bases refer to rewriting expressions so that the bases in the exponents are the same. This is particularly useful when solving equations involving exponents, as it simplifies comparison and manipulation. For example, numbers like 4, 16, and 8 can be rewritten using a common base of 2:

• \(4 = 2^2\)
• \(16 = 2^4\)
• \(8 = 2^3\)

Rewriting in this way unifies the bases and makes it easier to solve exponential equations by setting their exponents equal to each other when necessary. It’s essential to familiarize yourself with common bases, particularly powers of smaller integers like 2 or 3, as they frequently appear in problems involving exponential functions. This skill highlights the flexibility of using exponent rules to manipulate and solve equations, serving as a critical step toward finding solutions.

Solving Equations

Solving equations, especially those involving exponents, requires a systematic approach. Start by ensuring you express all terms with a common base if possible. This allows you to focus on the exponents alone. Once the bases are uniform,

• Combine like terms. If you have terms with similar bases and different exponents, they can often be combined using exponent addition or subtraction rules.
• Set the exponents equal to each other if the bases are the same, leading you to an equation purely in terms of the variable.
• Clear fractions by multiplying through by the denominator, eliminating fractional exponents which simplifies the solving process.

Once simplified, solve the resulting equation. If it leads to a quadratic form, apply the quadratic formula or other methods like factoring to find the solution. Remember that solving equations is fundamentally about isolating the variable, making sure to maintain balance (equality) throughout each step. This disciplined approach, using techniques like equating exponents, leads to efficient and accurate solutions.