Simplifying Exponential Expressions And Solving Equations

In this article, we will delve into the simplification of complex exponential expressions without relying on a calculator. Mastering these techniques is crucial for building a strong foundation in mathematics. We'll break down each step, providing detailed explanations and strategies to tackle these problems effectively. These exponential expressions often appear intimidating at first glance, but by understanding the fundamental rules of exponents and applying strategic simplification techniques, we can solve them with ease. The key is to break down the numbers into their prime factors and then apply the rules of exponents. Let's explore the simplification process for the following expressions:

2.1.1 Simplifying rac{10^{x+2} ullet 25^x ullet 8^x}{16{x+1} ullet 125^{x+1} ullet 2}

To simplify this expression, our primary goal is to express each term in its prime factors. This involves breaking down each number into its prime components (2, 3, 5, etc.) and then applying exponent rules to combine like terms. Understanding the properties of exponents is essential for this process, as it allows us to manipulate the expression strategically. We can rewrite the expression by expressing each term as a product of its prime factors raised to appropriate powers. Let's begin by breaking down each term:

• 10 = 2 * 5, so 10^(x+2) = (2 * 5)^(x+2) = 2^(x+2) * 5^(x+2)
• 25 = 5^2, so 25^x = (5²⁾x = 5^(2x)
• 8 = 2^3, so 8^x = (2³⁾x = 2^(3x)
• 16 = 2^4, so 16^(x+1) = (2⁴⁾(x+1) = 2^(4x+4)
• 125 = 5^3, so 125^(x+1) = (5³⁾(x+1) = 5^(3x+3)

Now, substitute these expressions back into the original equation:

rac{2^{x+2} ullet 5^{x+2} ullet 5^{2x} ullet 2^{3x}}{2{4x+4} ullet 5^{3x+3} ullet 2}

Next, we combine the terms with the same base by adding their exponents:

rac{2^{x+2+3x} ullet 5^{x+2+2x}}{2{4x+4+1} ullet 5^{3x+3}}

This simplifies to:

rac{2^{4x+2} ullet 5^{3x+2}}{2{4x+5} ullet 5^{3x+3}}

Now, we apply the rule for dividing exponents with the same base, which states that a^m / a^n = a^(m-n):

2^{(4x+2)-(4x+5)} ullet 5^{(3x+2)-(3x+3)}

This further simplifies to:

2^{-3} ullet 5^{-1}

Finally, we rewrite the expression with positive exponents:

rac{1}{2^3} ullet rac{1}{5} = rac{1}{8} ullet rac{1}{5} = rac{1}{40}

Therefore, the simplified form of the expression is 1/40. This detailed step-by-step simplification demonstrates the importance of breaking down complex expressions into manageable parts and applying the fundamental rules of exponents. Mastering these techniques will enable you to tackle more intricate problems with confidence.

2.1.2 Simplifying rac{3^{102} ullet 3^{101}}{3{204} - 6 ullet 3^{201}}

This expression requires careful manipulation and factoring to simplify effectively. The key here is to identify common factors and extract them to reduce the complexity of the expression. The expression involves both multiplication and subtraction, making it crucial to prioritize operations correctly. We begin by focusing on the numerator and denominator separately, identifying any potential simplifications within each part. Let's start by simplifying the numerator:

Simplifying the Numerator

The numerator is 3^{102} * 3^{101}. According to the product of powers rule, when multiplying exponents with the same base, we add the powers. Therefore:

3^{102} ullet 3^{101} = 3^{102+101} = 3^{203}

So, the numerator simplifies to 3^{203}.

Simplifying the Denominator

The denominator is 3^{204} - 6 * 3^{201}. To simplify this, we need to identify a common factor that can be extracted. Observe that 3^{201} is a common factor in both terms. We can rewrite the denominator by factoring out 3^{201}:

3^{204} - 6 ullet 3^{201} = 3^{201} ullet (3^3) - 6 ullet 3^{201}

Now, factor out 3^{201}:

3^{201} ullet (3^3 - 6)

Evaluate 3^3 which is 27:

3^{201} ullet (27 - 6)

This simplifies to:

3^{201} ullet 21

Combining the Simplified Numerator and Denominator

Now we have the simplified numerator and denominator:

rac{3^{203}}{3{201} ullet 21}

We can rewrite 21 as 3 * 7:

rac{3^{203}}{3{201} ullet 3 ullet 7}

Combine the 3 terms in the denominator:

rac{3^{203}}{3{202} ullet 7}

Now, apply the quotient of powers rule, which states that when dividing exponents with the same base, we subtract the powers:

3^{203-202} / 7

This simplifies to:

rac{3^1}{7} = rac{3}{7}

Therefore, the simplified form of the expression is 3/7. This example highlights the importance of factoring and recognizing common terms to simplify complex expressions. By extracting common factors and applying the rules of exponents, we can efficiently solve these types of problems.

Solving equations involving fractional exponents requires understanding how to manipulate these exponents and isolate the variable. In this section, we'll focus on solving equations of the form x^(m/n) = a without the aid of a calculator. The key to solving these equations is to raise both sides to the reciprocal power, which effectively cancels out the fractional exponent. This process involves several steps, including isolating the term with the fractional exponent, raising both sides to the appropriate power, and simplifying the resulting expression. Let's examine the following equation:

Solving for x in x^(2/3) = 4

To solve for x in the equation x^(2/3) = 4, we need to eliminate the fractional exponent. The fractional exponent is 2/3, so we raise both sides of the equation to the reciprocal power, which is 3/2. This is based on the principle that (x^(m/n))(n/m) = x. By applying this principle, we can isolate x and find its value. Let's proceed step-by-step:

Step 1: Raise Both Sides to the Reciprocal Power

Raise both sides of the equation x^(2/3) = 4 to the power of 3/2:

(x^{rac{2}{3}})^{rac{3}{2}} = 4^{rac{3}{2}}

Step 2: Simplify the Left Side

On the left side, the exponents multiply, and (2/3) * (3/2) = 1. Therefore:

x^1 = x

So, the left side simplifies to x.

Step 3: Simplify the Right Side

Now, we need to simplify 4^(3/2). This can be interpreted as (4^(1/2))3. First, find the square root of 4:

4^{rac{1}{2}} = sqr{4} = 2

Next, raise the result to the power of 3:

2^3 = 8

So, 4^(3/2) simplifies to 8.

Step 4: State the Solution

Now we have:

x = 8

Therefore, the solution to the equation x^(2/3) = 4 is x = 8. This detailed walkthrough demonstrates how to solve equations with fractional exponents by raising both sides to the reciprocal power. Understanding the properties of exponents and radicals is essential for mastering these types of problems. By following these steps, you can confidently solve similar equations without the use of a calculator.

This comprehensive guide has provided you with the tools and techniques necessary to simplify complex exponential expressions and solve equations with fractional exponents. By mastering these concepts, you will enhance your problem-solving skills and build a strong foundation in mathematics. Remember to practice regularly and apply these strategies to a variety of problems to solidify your understanding. The ability to simplify expressions and solve equations without a calculator is a valuable skill that will serve you well in various mathematical contexts.