Solving For P In Exponential Equations A Step By Step Guide

This article delves into the fascinating world of exponential equations, focusing on the critical skill of determining the value of the unknown exponent, denoted as 'p'. We'll dissect several equations, meticulously applying the fundamental laws of exponents to isolate 'p' and arrive at the correct solutions. This comprehensive guide is designed to enhance your understanding and proficiency in solving exponential equations, a crucial aspect of algebra and beyond.

Understanding the Fundamentals of Exponential Equations

Before we dive into the specific problems, let's solidify our understanding of the core principles governing exponential equations. At its heart, an exponential equation involves an unknown variable (in our case, 'p') residing in the exponent. Solving these equations hinges on the adept use of exponent rules, which dictate how exponents interact during multiplication, division, and exponentiation.

The Power of a Product Rule: A cornerstone of our toolkit is the rule that states when multiplying exponential expressions with the same base, we add the exponents. Mathematically, this is expressed as: a^m * aⁿ = a^m+n. This rule will be instrumental in simplifying expressions and combining terms.

Quotient of Powers Rule: Conversely, when dividing exponential expressions with the same base, we subtract the exponents: a^m / aⁿ = a^m-n. This rule allows us to reduce complex divisions into simpler forms.

Power of a Power Rule: When raising an exponential expression to another power, we multiply the exponents: (a^m)ⁿ = a^m*n. This rule is crucial for handling expressions where an exponent is itself raised to another power.

Equality of Exponents: A critical concept for solving for 'p' is the principle that if a^m = aⁿ, then m = n, provided that a is not equal to -1, 0, or 1. This allows us to equate the exponents once we've manipulated the equation to have the same base on both sides. This equality is the foundation for solving many exponential equations.

With these fundamental rules in mind, let's tackle the given problems step-by-step.

Problem 1: (-1)⁷ × (-1)⁹ = (-1)^p

This equation presents a classic application of the power of a product rule. Our immediate goal is to simplify the left-hand side of the equation by combining the exponential terms. We observe that both terms have the same base, which is -1. Therefore, we can apply the rule a^m * aⁿ = a^m+n.

Applying the power of a product rule, we add the exponents 7 and 9: (-1)⁷ × (-1)⁹ = (-1)⁷⁺⁹ = (-1)¹⁶. Now our equation transforms into (-1)¹⁶ = (-1)^p.

Now, we leverage the principle of equality of exponents. Since the bases on both sides of the equation are the same (-1), we can equate the exponents: 16 = p. Therefore, the solution to the equation is p = 16.

To solidify our understanding, let's verify the solution. Substituting p = 16 back into the original equation, we have (-1)⁷ × (-1)⁹ = (-1)¹⁶. We know that (-1) raised to an even power equals 1, so (-1)¹⁶ = 1. Also, (-1)⁷ = -1 and (-1)⁹ = -1, and (-1) × (-1) = 1. Thus, our solution p = 16 holds true.

Problem 2: (7/9)²¹ × (7/9)³ = (7/9)^3p

This problem, similar to the first, calls upon the power of a product rule. We have exponential terms with the same base (7/9) multiplied together. Our objective is to simplify the left-hand side and then solve for 'p'.

Applying the rule a^m * aⁿ = a^m+n, we combine the exponents on the left side: (7/9)²¹ × (7/9)³ = (7/9)²¹⁺³ = (7/9)²⁴. Now our equation becomes (7/9)²⁴ = (7/9)^3p.

Again, we employ the equality of exponents. The bases are identical on both sides, allowing us to equate the exponents: 24 = 3p. To isolate 'p', we divide both sides of the equation by 3: 24 / 3 = 3p / 3, which simplifies to 8 = p. Therefore, p = 8 is the solution.

To ensure accuracy, let's substitute p = 8 back into the original equation: (7/9)²¹ × (7/9)³ = (7/9)^3*8 = (7/9)²⁴. As we've already established, (7/9)²¹ × (7/9)³ indeed equals (7/9)²⁴, confirming the validity of our solution.

Problem 3: [(3/7)²]¹⁶ = (3/7)^p+5

This equation introduces the power of a power rule. On the left-hand side, we have an exponential term raised to another power. We need to simplify this using the rule (a^m)ⁿ = a^m*n before we can solve for 'p'.

Applying the power of a power rule, we multiply the exponents on the left side: [(3/7)²]¹⁶ = (3/7)^2*16 = (3/7)³². Our equation now looks like this: (3/7)³² = (3/7)^p+5.

The bases are the same on both sides, so we can utilize the equality of exponents: 32 = p + 5. To isolate 'p', we subtract 5 from both sides of the equation: 32 - 5 = p + 5 - 5, which gives us 27 = p. Thus, the solution is p = 27.

Verification is crucial. Substituting p = 27 back into the original equation, we get [(3/7)²]¹⁶ = (3/7)²⁷⁺⁵ = (3/7)³². As we previously calculated, [(3/7)²]¹⁶ does equal (3/7)³², confirming that our solution is correct.

Problem 4: (-2)¹³ ÷ (-2)¹¹ = (-2)^2p

This equation involves division of exponential terms with the same base, making the quotient of powers rule our tool of choice. We'll simplify the left-hand side and then solve for 'p'.

The quotient of powers rule states that a^m / aⁿ = a^m-n. Applying this to our equation, we get: (-2)¹³ ÷ (-2)¹¹ = (-2)^13-11 = (-2)². Now the equation is (-2)² = (-2)^2p.

We now apply the equality of exponents. Since the bases are the same, we equate the exponents: 2 = 2p. To isolate 'p', we divide both sides of the equation by 2: 2 / 2 = 2p / 2, resulting in 1 = p. Therefore, the solution is p = 1.

Let's verify our solution. Substituting p = 1 back into the original equation, we have (-2)¹³ ÷ (-2)¹¹ = (-2)^2*1 = (-2)². We already simplified the left side to (-2)², so our solution is confirmed.

Conclusion: Mastering Exponential Equations

Through these examples, we've demonstrated a systematic approach to solving exponential equations for 'p'. The key lies in a solid understanding of the fundamental rules of exponents: the power of a product rule, the quotient of powers rule, the power of a power rule, and the critical principle of equality of exponents. By applying these rules meticulously, we can simplify complex equations and isolate the unknown variable 'p'. Remember to always verify your solution by substituting it back into the original equation to ensure accuracy. With practice, solving exponential equations will become a comfortable and rewarding mathematical endeavor.

This article provided a comprehensive guide on solving exponential equations, focusing on finding the value of 'p'. By mastering these techniques, you'll be well-equipped to tackle a wide range of algebraic problems and further your mathematical expertise.