Simplifying Exponential Expressions A Step-by-Step Guide

In mathematics, simplifying complex expressions is a fundamental skill. This article provides a detailed guide on simplifying expressions involving exponents and roots, focusing on two intricate examples. We will break down each step, ensuring a clear understanding of the underlying principles and techniques. Whether you are a student looking to improve your algebra skills or a math enthusiast seeking to deepen your knowledge, this guide will provide valuable insights and practical strategies.

(i) Simplifying $(128)^{-2/7} - (625^{-3})^{-1/4} + 14(2401)^{-1/4}$

Exponential expressions can often appear daunting, but by understanding the properties of exponents and roots, we can systematically simplify them. This section will walk you through the step-by-step simplification of the expression $(128)^{-2/7} - (625^{-3})^{-1/4} + 14(2401)^{-1/4}$.

Step 1: Breaking Down the First Term $(128)^{-2/7}$

The first term we need to address is $(128)^{-2/7}$. The key here is to recognize that 128 is a power of 2. Specifically, $128 = 2^7$. This allows us to rewrite the term as $(2^7)^{-2/7}$.

Using the power of a power rule, which states that $(a^m)^n = a^{mn}$, we can simplify this further:

$(2^7)^{-2/7} = 2^{7 imes (-2/7)} = 2^{-2}$

Now, we have a negative exponent. Recall that a^{-n} = rac{1}{a^n}. Applying this rule:

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

So, the simplified form of $(128)^{-2/7}$ is rac{1}{4}. This initial step highlights the importance of recognizing powers and applying exponent rules to reduce complex expressions to simpler forms. By breaking down the number 128 into its prime factors, we were able to apply the power of a power rule and the negative exponent rule effectively. This approach is crucial for tackling similar exponential expressions.

Step 2: Simplifying the Second Term $(625^{-3})^{-1/4}$

Moving on to the second term, we have $(625^{-3})^{-1/4}$. Here, 625 is a power of 5, specifically $625 = 5^4$. Substituting this into the expression gives us $((5^4)^{-3})^{-1/4}$.

Again, we apply the power of a power rule twice. First:

$(5^4)^{-3} = 5^{4 imes -3} = 5^{-12}$

Now, we have $(5^{-12})^{-1/4}$. Applying the power of a power rule again:

$5^{-12 imes (-1/4)} = 5^{3}$

Finally, we calculate $5^3$:

$5^3 = 5 imes 5 imes 5 = 125$

Thus, $(625^{-3})^{-1/4}$ simplifies to 125. This step demonstrates how repeated application of the power of a power rule can simplify expressions with multiple exponents. Recognizing that 625 is a power of 5 was key to this simplification. The ability to identify such relationships is crucial in handling exponential expressions efficiently.

Step 3: Evaluating the Third Term $14(2401)^{-1/4}$

Now, let's focus on the third term: $14(2401)^{-1/4}$. We need to recognize that 2401 is a power of 7. In fact, $2401 = 7^4$. Substituting this into the expression, we get:

$14(7^4)^{-1/4}$

Applying the power of a power rule:

$14 imes 7^{4 imes (-1/4)} = 14 imes 7^{-1}$

Recall that 7^{-1} = rac{1}{7}, so we have:

14 imes rac{1}{7} = rac{14}{7} = 2

Therefore, $14(2401)^{-1/4}$ simplifies to 2. This step reinforces the importance of identifying powers and using the negative exponent rule. The constant factor 14 was multiplied after simplifying the exponential part, demonstrating a common strategy in simplifying expressions.

Step 4: Combining the Simplified Terms

Now that we have simplified each term individually, we can combine them:

(128)^{-2/7} - (625^{-3})^{-1/4} + 14(2401)^{-1/4} = rac{1}{4} - 125 + 2

To combine these terms, we need a common denominator for the fraction and the whole numbers. We can rewrite 2 as rac{8}{4} and -125 as rac{-500}{4}. Thus, the expression becomes:

rac{1}{4} - rac{500}{4} + rac{8}{4} = rac{1 - 500 + 8}{4} = rac{-491}{4}

So, the final simplified form of the expression is -rac{491}{4}. This final step involves basic arithmetic operations, highlighting that simplification often reduces to combining like terms after applying exponent and root rules. The result, -rac{491}{4}, is the simplified form of the original complex expression.

(ii) Simplifying rac{4}{(2187)^{-3/7}} - rac{5}{(256)^{-1/4}} + rac{2}{(1331)^{-1/3}}

This section delves into simplifying another complex expression: rac{4}{(2187)^{-3/7}} - rac{5}{(256)^{-1/4}} + rac{2}{(1331)^{-1/3}}. This expression involves fractions with exponents in the denominator, which requires careful application of exponent rules and simplification techniques. Let's break down the simplification process step by step.

Step 1: Simplifying the First Term rac{4}{(2187)^{-3/7}}

The first term we need to simplify is rac{4}{(2187)^{-3/7}}. To simplify this, we first recognize that 2187 is a power of 3. Specifically, $2187 = 3^7$. Substituting this into the expression, we have:

rac{4}{(3^7)^{-3/7}}

Using the power of a power rule, which states $(a^m)^n = a^{mn}$, we get:

rac{4}{3^{7 imes (-3/7)}} = rac{4}{3^{-3}}

Now, we have a negative exponent in the denominator. Recall that a^{-n} = rac{1}{a^n}, so rac{1}{a^{-n}} = a^n. Applying this rule, we get:

$4 imes 3^3$

Calculating $3^3$ gives us $3 imes 3 imes 3 = 27$. Thus, the term becomes:

$4 imes 27 = 108$

So, the simplified form of rac{4}{(2187)^{-3/7}} is 108. This step highlights how recognizing powers and applying exponent rules can transform a complex fraction into a simple integer. The transformation of the negative exponent in the denominator is a crucial technique in simplifying such expressions.

Step 2: Simplifying the Second Term rac{5}{(256)^{-1/4}}

Moving on to the second term, we have rac{5}{(256)^{-1/4}}. We need to identify that 256 is a power of 4 (or 2), specifically $256 = 4^4$ (or $2^8$). Using $256 = 4^4$, the expression becomes:

rac{5}{(4^4)^{-1/4}}

Applying the power of a power rule:

rac{5}{4^{4 imes (-1/4)}} = rac{5}{4^{-1}}

Again, we have a negative exponent in the denominator. Using the rule rac{1}{a^{-n}} = a^n, we get:

$5 imes 4^1 = 5 imes 4 = 20$

Thus, rac{5}{(256)^{-1/4}} simplifies to 20. This step reinforces the technique of dealing with negative exponents in the denominator and showcases the simplification process when the base is a power of a simple number. Recognizing 256 as $4^4$ allowed for a straightforward application of the exponent rules.

Step 3: Simplifying the Third Term rac{2}{(1331)^{-1/3}}

Now, let's simplify the third term: rac{2}{(1331)^{-1/3}}. We need to recognize that 1331 is a power of 11. In fact, $1331 = 11^3$. Substituting this into the expression, we get:

rac{2}{(11^3)^{-1/3}}

Applying the power of a power rule:

rac{2}{11^{3 imes (-1/3)}} = rac{2}{11^{-1}}

Using the rule rac{1}{a^{-n}} = a^n, we have:

$2 imes 11^1 = 2 imes 11 = 22$

Therefore, rac{2}{(1331)^{-1/3}} simplifies to 22. This step further demonstrates the effectiveness of identifying powers and using the negative exponent rule. Recognizing 1331 as $11^3$ was crucial for simplifying this term.

Step 4: Combining the Simplified Terms

Now that we have simplified each term individually, we can combine them:

rac{4}{(2187)^{-3/7}} - rac{5}{(256)^{-1/4}} + rac{2}{(1331)^{-1/3}} = 108 - 20 + 22

Performing the arithmetic operations:

$108 - 20 + 22 = 88 + 22 = 110$

So, the final simplified form of the expression is 110. This final step involves simple addition and subtraction, highlighting that complex expressions often reduce to basic arithmetic after applying the necessary algebraic simplifications. The result, 110, is the simplified value of the original complex expression.

Conclusion

In conclusion, simplifying complex expressions involving exponents and roots requires a solid understanding of exponent rules and the ability to recognize powers of numbers. By breaking down each term step by step, applying the appropriate rules, and simplifying, we can reduce even the most daunting expressions to manageable forms. The examples discussed in this article provide a comprehensive guide to these techniques, enabling you to tackle similar problems with confidence. Mastering these skills is crucial for success in algebra and beyond. Remember, the key is to identify powers, apply exponent rules systematically, and combine like terms to arrive at the simplest form of the expression.