Simplifying Exponential Expressions A Step-by-Step Guide

This article delves into the fascinating realm of simplifying exponential expressions, a fundamental concept in mathematics. We will explore various problems involving exponents and provide step-by-step solutions, equipping you with the knowledge and skills to tackle complex expressions with confidence. Understanding exponential expressions is crucial for success in algebra, calculus, and other advanced mathematical topics. Mastering the rules of exponents not only simplifies calculations but also provides a deeper understanding of mathematical relationships. This article aims to break down complex problems into manageable steps, ensuring a clear and concise learning experience for all readers.

Problem 3: Simplifying ${\frac{7^{14} \cdot (7^{2})^{3}}{(7^{3})^{6} \cdot 7^{2}}}$

Let's begin by simplifying the expression ${\frac{7^{14} \cdot (7^{2})^{3}}{(7^{3})^{6} \cdot 7^{2}}}$ . This problem involves several properties of exponents, which we will apply systematically. Our primary goal is to reduce the expression to its simplest form, utilizing the rules of exponents. The key to solving this problem lies in understanding how exponents interact with each other during multiplication, division, and raising a power to a power. This step-by-step approach will not only help in solving this particular problem but also in developing a broader understanding of exponential simplification.

Step 1: Apply the Power of a Power Rule

The power of a power rule states that ${(a^{m})^{n} = a^{m \cdot n}}$. Applying this rule to the numerator, we have

${ (7^{2})^{3} = 7^{2 \cdot 3} = 7^{6} }$

Similarly, in the denominator,

${ (7^{3})^{6} = 7^{3 \cdot 6} = 7^{18} }$

This step is crucial as it simplifies the complex exponents, making the expression easier to manage. The power of a power rule is a cornerstone in simplifying exponential expressions, and mastering it is essential for tackling more complex problems. By applying this rule, we transform the original expression into a more workable form.

Step 2: Substitute the Simplified Terms

Now, substitute these simplified terms back into the original expression:

${ \frac{7^{14} \cdot 7^{6}}{7^{18} \cdot 7^{2}} }$

This substitution allows us to rewrite the expression with simplified exponents, bringing us closer to the final solution. By replacing the original terms with their simplified counterparts, we create a clearer picture of the expression's structure. This step is a bridge between the initial complex form and the subsequent application of further exponent rules.

Step 3: Apply the Product of Powers Rule

The product of powers rule states that ${a^{m} \cdot a^{n} = a^{m+n}}$. Applying this rule to both the numerator and the denominator:

Numerator:

${ 7^{14} \cdot 7^{6} = 7^{14+6} = 7^{20} }$

Denominator:

${ 7^{18} \cdot 7^{2} = 7^{18+2} = 7^{20} }$

This step consolidates the exponents in both the numerator and the denominator, making the expression even simpler. The product of powers rule is fundamental in combining terms with the same base, and its application here significantly reduces the complexity of the expression.

Step 4: Substitute the Simplified Terms Again

Substitute these results back into the expression:

${ \frac{7^{20}}{7^{20}} }$

This substitution further simplifies the expression, revealing a clear path to the final solution. By replacing the previous terms with their consolidated forms, we highlight the underlying structure of the expression and set the stage for the final simplification.

Step 5: Apply the Quotient of Powers Rule

The quotient of powers rule states that ${\frac{a^{m}}{a^{n}} = a^{m-n}}$. Applying this rule:

${ \frac{7^{20}}{7^{20}} = 7^{20-20} = 7^{0} }$

This step uses the quotient of powers rule to simplify the fraction, leading us to the final exponent value. The quotient of powers rule is crucial for dividing terms with the same base, and its application here results in a significant simplification of the expression.

Step 6: Simplify the Expression

Finally, any non-zero number raised to the power of 0 is 1. Therefore,

${ 7^{0} = 1 }$

This final step completes the simplification, providing the ultimate solution to the problem. The rule that any non-zero number raised to the power of 0 equals 1 is a fundamental concept in exponents, and its application here finalizes the simplification process.

Final Answer

The simplified form of the expression ${\frac{7^{14} \cdot (7^{2})^{3}}{(7^{3})^{6} \cdot 7^{2}}}$ is:

${ 1 }$

Problem 4: Simplifying ${\frac{25^{3} \cdot 125^{2}}{5^{10}}}$

Now, let's tackle the expression ${\frac{25^{3} \cdot 125^{2}}{5^{10}}}$ . This problem requires us to express all terms with a common base, which in this case is 5. By converting all terms to the same base, we can effectively apply the rules of exponents to simplify the expression. This approach is a common strategy when dealing with exponential expressions involving different bases that are powers of the same number.

Step 1: Express all Terms with the Same Base

Rewrite 25 and 125 as powers of 5:

${ 25 = 5^{2} }$

${ 125 = 5^{3} }$

This step is crucial as it sets the foundation for applying the rules of exponents. By expressing all terms with the same base, we create a unified structure that allows for straightforward simplification. Recognizing the common base is a key skill in solving exponential problems.

Step 2: Substitute the Equivalent Terms

Substitute these expressions back into the original equation:

${ \frac{(5^{2})^{3} \cdot (5^{3})^{2}}{5^{10}} }$

This substitution allows us to rewrite the expression in terms of a single base, making it easier to simplify. By replacing the original terms with their equivalent forms in base 5, we create a more manageable expression that is ready for further simplification using exponent rules.

Step 3: Apply the Power of a Power Rule

Apply the power of a power rule, ${(a^{m})^{n} = a^{m \cdot n}}$, to both terms in the numerator:

${ (5^{2})^{3} = 5^{2 \cdot 3} = 5^{6} }$

${ (5^{3})^{2} = 5^{3 \cdot 2} = 5^{6} }$

This step simplifies the exponents in the numerator, bringing us closer to a simplified expression. The power of a power rule is instrumental in reducing the complexity of exponential expressions, and its application here is a key step in the simplification process.

Step 4: Substitute the Simplified Terms

Substitute these results back into the expression:

${ \frac{5^{6} \cdot 5^{6}}{5^{10}} }$

This substitution further simplifies the expression, making it easier to apply the remaining exponent rules. By replacing the previous terms with their simplified forms, we create a clearer picture of the expression's structure and pave the way for the next steps in the simplification process.

Step 5: Apply the Product of Powers Rule

Apply the product of powers rule, ${a^{m} \cdot a^{n} = a^{m+n}}$, to the numerator:

${ 5^{6} \cdot 5^{6} = 5^{6+6} = 5^{12} }$

This step combines the exponents in the numerator, further simplifying the expression. The product of powers rule is a fundamental tool for combining terms with the same base, and its application here significantly reduces the complexity of the numerator.

Step 6: Substitute the Simplified Term

Substitute the result back into the expression:

${ \frac{5^{12}}{5^{10}} }$

This substitution simplifies the expression even further, setting the stage for the final simplification using the quotient of powers rule. By replacing the previous terms with their consolidated forms, we create a more streamlined expression that is ready for the final steps in the simplification process.

Step 7: Apply the Quotient of Powers Rule

Apply the quotient of powers rule, ${\frac{a^{m}}{a^{n}} = a^{m-n}}$:

${ \frac{5^{12}}{5^{10}} = 5^{12-10} = 5^{2} }$

This step uses the quotient of powers rule to simplify the fraction, leading us to the final exponent value. The quotient of powers rule is crucial for dividing terms with the same base, and its application here results in a significant simplification of the expression.

Step 8: Simplify the Expression

Finally, simplify ${5^{2}}$:

${ 5^{2} = 25 }$

This final step completes the simplification, providing the ultimate solution to the problem. Evaluating the final exponent yields the simplified value of the expression.

Final Answer

The simplified form of the expression ${\frac{25^{3} \cdot 125^{2}}{5^{10}}}$ is:

${ 25 }$

Problem 5: Simplifying ${\frac{3^{8} \cdot 7^{8}}{21^{7}}}$

Now, let's consider the expression ${\frac{3^{8} \cdot 7^{8}}{21^{7}}}$ . To simplify this, we will express the denominator in terms of its prime factors, which are 3 and 7. This allows us to use the properties of exponents to combine and simplify the expression. Breaking down the denominator into its prime factors is a crucial step in simplifying expressions involving different bases.

Step 1: Express the Denominator in Terms of Prime Factors

Rewrite 21 as a product of its prime factors:

${ 21 = 3 \cdot 7 }$

This step is crucial as it allows us to express the denominator in terms of the same bases as the numerator. By breaking down 21 into its prime factors, we create a common ground for applying the rules of exponents.

Step 2: Substitute the Equivalent Term

Substitute this expression back into the original equation:

${ \frac{3^{8} \cdot 7^{8}}{(3 \cdot 7)^{7}} }$

This substitution allows us to rewrite the expression with a denominator that is expressed in terms of its prime factors. By replacing 21 with its prime factorization, we create a more manageable expression that is ready for further simplification using exponent rules.

Step 3: Apply the Power of a Product Rule

Apply the power of a product rule, ${(ab)^{n} = a^{n}b^{n}}$, to the denominator:

${ (3 \cdot 7)^{7} = 3^{7} \cdot 7^{7} }$

This step simplifies the denominator by distributing the exponent to each factor. The power of a product rule is essential for simplifying expressions where a product is raised to a power, and its application here is a key step in the simplification process.

Step 4: Substitute the Simplified Term

Substitute this result back into the expression:

${ \frac{3^{8} \cdot 7^{8}}{3^{7} \cdot 7^{7}} }$

This substitution further simplifies the expression, making it easier to apply the remaining exponent rules. By replacing the denominator with its simplified form, we create a clearer picture of the expression's structure and pave the way for the next steps in the simplification process.

Step 5: Apply the Quotient of Powers Rule

Apply the quotient of powers rule, ${\frac{a^{m}}{a^{n}} = a^{m-n}}$, separately for the bases 3 and 7:

For base 3:

${ \frac{3^{8}}{3^{7}} = 3^{8-7} = 3^{1} = 3 }$

For base 7:

${ \frac{7^{8}}{7^{7}} = 7^{8-7} = 7^{1} = 7 }$

This step uses the quotient of powers rule to simplify the expression for each base. The quotient of powers rule is crucial for dividing terms with the same base, and its application here results in a significant simplification of the expression.

Step 6: Combine the Results

Multiply the simplified terms:

${ 3 \cdot 7 = 21 }$

This final step combines the simplified terms, providing the ultimate solution to the problem. Multiplying the simplified values for each base completes the simplification process.

Final Answer

The simplified form of the expression ${\frac{3^{8} \cdot 7^{8}}{21^{7}}}$ is:

${ 21 }$

Problem 6: Simplifying ${\frac{5^{9} \cdot 4^{6}}{20^{6}}}$

Finally, let's simplify the expression ${\frac{5^{9} \cdot 4^{6}}{20^{6}}}$ . Similar to the previous problem, we will express all terms in terms of their prime factors. This approach will allow us to apply the rules of exponents effectively. Expressing all terms in terms of their prime factors is a common and powerful strategy for simplifying exponential expressions.

Step 1: Express all Terms in Terms of Prime Factors

Rewrite 4 and 20 as products of their prime factors:

${ 4 = 2^{2} }$

${ 20 = 2^{2} \cdot 5 }$

This step is crucial as it sets the stage for applying the rules of exponents. By expressing all terms in terms of their prime factors, we create a common ground for simplification.

Step 2: Substitute the Equivalent Terms

Substitute these expressions back into the original equation:

${ \frac{5^{9} \cdot (2^{2})^{6}}{(2^{2} \cdot 5)^{6}} }$

This substitution allows us to rewrite the expression in terms of its prime factors, making it easier to simplify. By replacing the original terms with their prime factorizations, we create a more manageable expression that is ready for further simplification using exponent rules.

Step 3: Apply the Power of a Power Rule

Apply the power of a power rule, ${(a^{m})^{n} = a^{m \cdot n}}$, to the term in the numerator:

${ (2^{2})^{6} = 2^{2 \cdot 6} = 2^{12} }$

This step simplifies the exponent in the numerator, bringing us closer to a simplified expression. The power of a power rule is instrumental in reducing the complexity of exponential expressions, and its application here is a key step in the simplification process.

Step 4: Substitute the Simplified Term

Substitute this result back into the expression:

${ \frac{5^{9} \cdot 2^{12}}{(2^{2} \cdot 5)^{6}} }$

This substitution further simplifies the expression, making it easier to apply the remaining exponent rules. By replacing the previous term with its simplified form, we create a clearer picture of the expression's structure and pave the way for the next steps in the simplification process.

Step 5: Apply the Power of a Product Rule

Apply the power of a product rule, ${(ab)^{n} = a^{n}b^{n}}$, to the denominator:

${ (2^{2} \cdot 5)^{6} = (2^{2})^{6} \cdot 5^{6} }$

This step simplifies the denominator by distributing the exponent to each factor. The power of a product rule is essential for simplifying expressions where a product is raised to a power, and its application here is a key step in the simplification process.

Step 6: Apply the Power of a Power Rule Again

Apply the power of a power rule, ${(a^{m})^{n} = a^{m \cdot n}}$, to the term in the denominator:

${ (2^{2})^{6} = 2^{2 \cdot 6} = 2^{12} }$

This step further simplifies the denominator, bringing us closer to a simplified expression. The power of a power rule is instrumental in reducing the complexity of exponential expressions, and its application here is a key step in the simplification process.

Step 7: Substitute the Simplified Terms

Substitute these results back into the expression:

${ \frac{5^{9} \cdot 2^{12}}{2^{12} \cdot 5^{6}} }$

This substitution further simplifies the expression, making it easier to apply the remaining exponent rules. By replacing the previous terms with their simplified forms, we create a clearer picture of the expression's structure and pave the way for the next steps in the simplification process.

Step 8: Apply the Quotient of Powers Rule

Apply the quotient of powers rule, ${\frac{a^{m}}{a^{n}} = a^{m-n}}$, separately for the bases 5 and 2:

For base 5:

${ \frac{5^{9}}{5^{6}} = 5^{9-6} = 5^{3} }$

For base 2:

${ \frac{2^{12}}{2^{12}} = 2^{12-12} = 2^{0} = 1 }$

This step uses the quotient of powers rule to simplify the expression for each base. The quotient of powers rule is crucial for dividing terms with the same base, and its application here results in a significant simplification of the expression.

Step 9: Simplify the Expression

Simplify ${5^{3}}$:

${ 5^{3} = 5 \cdot 5 \cdot 5 = 125 }$

This final step completes the simplification, providing the ultimate solution to the problem. Evaluating the final exponent yields the simplified value of the expression.

Final Answer

The simplified form of the expression ${\frac{5^{9} \cdot 4^{6}}{20^{6}}}$ is:

${ 125 }$

Conclusion

In this article, we have thoroughly explored the simplification of exponential expressions through a series of detailed examples. By applying the fundamental rules of exponents, such as the power of a power rule, the product of powers rule, and the quotient of powers rule, we have successfully simplified complex expressions. Each problem demonstrated a unique approach, reinforcing the importance of understanding the underlying principles. Mastering these techniques is crucial for advanced mathematical studies and real-world applications. We encourage you to practice these methods and apply them to various problems to strengthen your understanding and proficiency in simplifying exponential expressions. The journey through these problems not only enhances your mathematical skills but also cultivates a deeper appreciation for the elegance and power of exponential notation.