Simplifying Expressions A Step By Step Guide To Evaluating (4^3) / (0.5^6 * 32^2)

JU07/16/2025 08, 2025 by THE IDEN 82 views

$Simplify or Evaluate the Expression ${ \frac{4^3}{0.5^6 \cdot 32^2} }$$

Understanding the Expression

To simplify or evaluate the expression ${ \frac{4^3}{0.5^6 \cdot 32^2} }$ , we need to break it down into its components and apply the rules of exponents and fractions. This involves understanding how to represent numbers in different forms, such as powers of 2, and how to manipulate these powers to simplify the overall expression. The key is to rewrite each term in a common base, which in this case is 2, allowing us to combine the exponents and simplify the expression efficiently. Our main goal is to arrive at a simplified numerical value by systematically reducing the complexity of the given fraction. Let's embark on this mathematical journey together, exploring each step with clarity and precision, ensuring a thorough understanding of the underlying principles.

The numerator, ${4^3}$ , can be expressed as ${(2^2)^3}$ , which simplifies to ${2^6}$ using the power of a power rule. The denominator consists of two terms: ${0.5^6}$ and ${32^2}$ . We can rewrite 0.5 as ${\frac{1}{2}}$ or ${2^{-1}}$ , so ${0.5^6}$ becomes ${(2^{-1})^6}$ , which simplifies to ${2^{-6}}$ . Next, 32 is a power of 2, specifically ${2^5}$ , so ${32^2}$ can be written as ${(2^5)^2}$ , which simplifies to ${2^{10}}$ . Now, let's combine these simplified terms back into the original expression. We have ${\frac{2^6}{2^{-6} \cdot 2^{10}}}$ . To further simplify, we multiply the terms in the denominator, which means adding their exponents: ${2^{-6} \cdot 2^{10} = 2^{-6+10} = 2^4}$ . The expression now looks like ${\frac{2^6}{2^4}}$ . Finally, to divide terms with the same base, we subtract the exponents: ${2^{6-4} = 2^2}$ . Thus, the expression simplifies to ${2^2}$ , which equals 4. This step-by-step simplification not only arrives at the correct answer but also illuminates the path, making the process understandable and replicable for similar problems. The essence of this mathematical simplification lies in the adept use of exponent rules and the ability to represent numbers in their prime factor forms, facilitating a seamless transition from a complex expression to a simple numerical value.

The final step in simplifying or evaluating the expression involves calculating the value of ${2^2}$ , which is simply 4. This result underscores the power of simplifying complex expressions into manageable forms. By breaking down each component of the original expression, applying exponent rules, and then combining like terms, we've successfully transformed a seemingly intricate problem into a straightforward calculation. This approach not only provides the answer but also deepens our understanding of the mathematical principles at play. The beauty of this simplification process lies in its systematic nature, allowing for complex problems to be tackled with confidence and precision. Furthermore, the ability to convert numbers into their exponential forms is a fundamental skill in mathematics, enabling us to manipulate expressions more effectively. The journey from the initial expression to the final numerical value of 4 highlights the elegance and efficiency of mathematical simplification techniques, reinforcing the idea that even the most daunting expressions can be tamed with the right approach and understanding.

Step-by-Step Breakdown

Rewrite 4 as a power of 2: ${4 = 2^2}$ . Therefore, ${4^3 = (2^2)^3 = 2^6}$ . This transformation is crucial as it sets the stage for expressing all terms in the expression using the same base, which is a prerequisite for applying the exponent rules effectively. The ability to recognize and utilize such equivalencies is a hallmark of mathematical proficiency, enabling the simplification of complex expressions into more manageable forms. By expressing 4 as a power of 2, we lay the groundwork for a seamless simplification process, turning what might initially appear as a daunting expression into a solvable equation. This step is not merely a cosmetic change; it is a strategic maneuver that unlocks the potential for further simplification, underscoring the importance of strategic thinking in mathematics.
Rewrite 0.5 as a power of 2: ${0.5 = \frac{1}{2} = 2^{-1}}$ . Thus, ${0.5^6 = (2^{-1})^6 = 2^{-6}}$ . Converting decimal fractions into exponential forms is a pivotal step in simplifying expressions involving both decimals and exponents. By expressing 0.5 as ${2^{-1}}$ , we facilitate the application of exponent rules, making the subsequent steps in the simplification process more straightforward. This conversion not only bridges the gap between decimal and exponential representations but also showcases the versatility of exponential notation in handling various numerical formats. The elegance of this step lies in its ability to transform a potentially cumbersome decimal into a manageable exponential term, exemplifying the power of mathematical transformations in problem-solving.
Rewrite 32 as a power of 2: ${32 = 2^5}$ . So, ${32^2 = (2^5)^2 = 2^{10}}$ . Recognizing powers of 2 is a fundamental skill in mathematics, and expressing 32 as ${2^5}$ is a critical step in simplifying our expression. This transformation allows us to maintain a consistent base throughout the equation, which is essential for applying exponent rules correctly. By rewriting 32 in its exponential form, we not only simplify the term itself but also align it with the other terms in the expression, paving the way for further simplification. This step highlights the importance of pattern recognition and the ability to identify common bases in mathematical expressions, enabling us to streamline the simplification process and arrive at a solution more efficiently.
Substitute the simplified terms back into the expression: The expression becomes ${ \frac{2^6}{2^{-6} \cdot 2^{10}} }$ . With each term now expressed as a power of 2, we're poised to apply the rules of exponents to simplify the expression further. This substitution is not just a mechanical step; it's a pivotal moment where the strategic transformations made earlier come to fruition. By replacing the original terms with their simplified exponential forms, we've set the stage for a streamlined simplification process. The clarity and elegance of this step underscore the power of strategic problem-solving in mathematics, where each transformation is carefully chosen to pave the way for subsequent simplifications.
Simplify the denominator: ${2^{-6} \cdot 2^{10} = 2^{-6+10} = 2^4}$ . When multiplying terms with the same base, we add their exponents. This is a fundamental rule of exponents that simplifies the expression significantly. By adding the exponents -6 and 10, we consolidate the denominator into a single term, ${2^4}$ . This step not only simplifies the expression but also showcases the elegance and efficiency of exponent rules in mathematical manipulations. The ability to apply these rules correctly is a cornerstone of algebraic simplification, allowing us to transform complex expressions into more manageable forms with ease.
Simplify the entire expression: ${ \frac{2^6}{2^4} = 2^{6-4} = 2^2 }$ . Dividing terms with the same base involves subtracting the exponents. This fundamental rule allows us to further simplify the expression by reducing it to a single exponential term. By subtracting the exponent in the denominator (4) from the exponent in the numerator (6), we arrive at ${2^2}$ . This step exemplifies the power of exponent rules in streamlining mathematical expressions, making complex calculations more manageable and revealing the underlying simplicity of the equation.
Evaluate the final power: ${2^2 = 4}$ . The final step in evaluating the expression is to calculate the value of ${2^2}$ , which is 4. This simple calculation completes the simplification process, providing a clear and concise answer to the original problem. The journey from the initial complex expression to the final numerical value of 4 underscores the effectiveness of strategic simplification techniques in mathematics. By breaking down the problem into manageable steps and applying the rules of exponents, we've successfully transformed a seemingly intricate expression into a straightforward calculation. This result not only answers the question but also reinforces the beauty and efficiency of mathematical problem-solving.

Conclusion

In conclusion, by systematically applying the rules of exponents and simplifying each term, we have successfully evaluated the expression ${ \frac{4^3}{0.5^6 \cdot 32^2} }$ to be 4. This exercise highlights the importance of understanding and applying fundamental mathematical principles to solve complex problems. The journey from the initial expression to the final answer demonstrates the power of simplification techniques and the elegance of mathematical reasoning. By breaking down the problem into smaller, manageable steps, we've not only found the solution but also gained a deeper appreciation for the underlying mathematical concepts. This approach is invaluable for tackling a wide range of mathematical challenges, empowering us to approach complex problems with confidence and precision.