Equivalent Expression For ((3/4)^4 (3/4)^3)^2

Introduction

In the realm of mathematics, simplifying expressions is a fundamental skill. This article delves into the process of finding an equivalent expression for the given mathematical expression: $\left(\left(\frac{3}{4}\right)^4\left(\frac{3}{4}\right)^3\right)^2$. We will explore the underlying principles of exponents and apply them step-by-step to arrive at the correct answer. Understanding these concepts is crucial for students and anyone dealing with mathematical problems involving exponents.

Understanding the Expression

Our initial expression is $\left(\left(\frac{3}{4}\right)^4\left(\frac{3}{4}\right)^3\right)^2$. This may look daunting at first, but we can break it down using the laws of exponents. The key here is to recognize that we are dealing with powers of the same base, which is $\frac{3}{4}$.

To effectively simplify this expression, it is essential to have a firm grasp of the rules of exponents. These rules allow us to manipulate and simplify expressions involving powers. One of the primary rules we will use is the product of powers rule, which states that when multiplying expressions with the same base, you add the exponents. Another crucial rule is the power of a power rule, which states that when raising a power to another power, you multiply the exponents. These rules are the bedrock of simplifying expressions like the one we are tackling.

The expression consists of several layers, each requiring careful attention. First, we have the inner part, which involves multiplying two terms with the same base. Then, we have an outer power that applies to the result of the inner operation. This nested structure requires us to work from the inside out, ensuring we correctly apply the rules of exponents at each stage. By dissecting the expression in this way, we can systematically simplify it and arrive at the correct equivalent form.

Let's begin by addressing the inner part of the expression. We have $\left(\frac{3}{4}\right)^4$ multiplied by $\left(\frac{3}{4}\right)^3$. Both terms have the same base, $\frac{3}{4}$, which allows us to use the product of powers rule. This rule simplifies the expression significantly and sets the stage for the next step in our simplification process. By understanding the structure of the expression and the rules that govern it, we can approach the problem with confidence and accuracy.

Step-by-Step Simplification

Step 1: Simplifying the Inner Expression

We start by simplifying the expression inside the parentheses: $\left(\frac{3}{4}\right)^4\left(\frac{3}{4}\right)^3$. According to the product of powers rule, when multiplying powers with the same base, we add the exponents. In this case, the base is $\frac{3}{4}$, and the exponents are 4 and 3. Therefore, we add the exponents: 4 + 3 = 7. This gives us:

$$\left(\frac{3}{4}\right)^4\left(\frac{3}{4}\right)^3 = \left(\frac{3}{4}\right)^{4+3} = \left(\frac{3}{4}\right)^7$$

This step is crucial as it simplifies the expression significantly. By applying the product of powers rule, we have reduced the two terms into a single term with a combined exponent. This simplification makes the subsequent steps more manageable and less prone to errors. It is a clear example of how understanding and applying the fundamental rules of exponents can streamline mathematical problem-solving. The resulting expression, $\left(\frac{3}{4}\right)^7$, is now much easier to handle in the next stage of simplification.

Step 2: Applying the Outer Exponent

Now we have $\left(\left(\frac{3}{4}\right)^7\right)^2$. Here, we apply the power of a power rule, which states that when raising a power to another power, we multiply the exponents. In this case, we have $\left(\frac{3}{4}\right)^7$ raised to the power of 2. So, we multiply the exponents: 7 * 2 = 14. This gives us:

$$\left(\left(\frac{3}{4}\right)^7\right)^2 = \left(\frac{3}{4}\right)^{7\times 2} = \left(\frac{3}{4}\right)^{14}$$

This step completes the simplification process. By applying the power of a power rule, we have transformed the expression into its simplest form. The exponent of 14 represents the combined effect of both the inner and outer exponents. This result demonstrates the power and efficiency of using exponent rules to simplify complex expressions. Understanding this process is vital for solving more intricate mathematical problems involving powers and exponents. The final simplified expression, $\left(\frac{3}{4}\right)^{14}$, is now ready to be compared with the given options.

Analyzing the Options

We have simplified the original expression to $\left(\frac{3}{4}\right)^{14}$. Now, let's compare this result with the given options:

• A. $\left(\frac{9}{16}\right)^{24}$
• B. $\left(\frac{9}{16}\right)^{14}$
• C. $\left(\frac{3}{4}\right)^{24}$
• D. $\left(\frac{3}{4}\right)^{14}$

Upon comparing our simplified expression with the options, it is evident that option D, $\left(\frac{3}{4}\right)^{14}$, matches our result perfectly. This confirms that we have correctly applied the rules of exponents and arrived at the correct equivalent expression. The other options can be ruled out as they do not match the simplified form we obtained.

Option A involves a different base and exponent, making it clearly incorrect. Option B has the correct base when $\frac{3}{4}$ is squared to get $\frac{9}{16}$, but the exponent is incorrect. Option C has the correct base but an incorrect exponent. Only option D aligns precisely with our simplified expression, validating our step-by-step simplification process and highlighting the importance of accuracy in applying mathematical rules.

Conclusion

After simplifying the expression $\left(\left(\frac{3}{4}\right)^4\left(\frac{3}{4}\right)^3\right)^2$, we found that the equivalent expression is $\left(\frac{3}{4}\right)^{14}$. This corresponds to option D. The process involved using the product of powers rule and the power of a power rule, which are fundamental concepts in dealing with exponents. Mastering these rules is essential for simplifying complex mathematical expressions efficiently and accurately.

This exercise underscores the significance of understanding and applying the rules of exponents. By breaking down the expression into manageable steps, we were able to systematically simplify it and arrive at the correct answer. This approach not only helps in solving mathematical problems but also enhances problem-solving skills in general. The ability to dissect complex problems into simpler components is a valuable asset in various fields, not just mathematics.

In summary, the key to simplifying expressions involving exponents lies in recognizing the underlying rules and applying them methodically. In this case, the product of powers rule allowed us to combine terms with the same base, and the power of a power rule enabled us to handle nested exponents. By mastering these techniques, students and professionals alike can confidently tackle a wide range of mathematical challenges. The correct answer, $\left(\frac{3}{4}\right)^{14}$, serves as a testament to the effectiveness of these methods and the importance of precision in mathematical calculations.