Simplify The Expression: $\[ \left(\frac{2}{5}\right)^4 \cdot \left(\frac{4}{25}\right)^3 \\]

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. When dealing with fractions and exponents, it's essential to understand the rules of exponentiation and how to manipulate expressions to simplify them. In this article, we will focus on simplifying the given expression: ${ \left(\frac{2}{5}\right)^4 \cdot \left(\frac{4}{25}\right)^3 }$. We will break down the steps involved in simplifying this expression and provide a clear explanation of the mathematical concepts used.

Understanding Exponentiation

Before we dive into simplifying the expression, let's review the rules of exponentiation. When a fraction is raised to a power, we raise both the numerator and the denominator to that power. For example, ${ \left(\frac{2}{5}\right)^4 }$ means ${ \frac{24}{54} }$. Similarly, ${ \left(\frac{4}{25}\right)^3 }$ means ${ \frac{43}{253} }$.

Simplifying the Expression

Now that we understand the rules of exponentiation, let's simplify the given expression. We can start by simplifying each fraction separately.

(25)4â‹…(425)3{ \left(\frac{2}{5}\right)^4 \cdot \left(\frac{4}{25}\right)^3 }

$= 2454â‹…43253{ \frac{2^4}{5^4} \cdot \frac{4^3}{25^3} }$

$= 24â‹…4354â‹…253{ \frac{2^4 \cdot 4^3}{5^4 \cdot 25^3} }$

Simplifying the Numerator

To simplify the numerator, we need to calculate the values of 242^4 and 434^3. We know that 24=162^4 = 16 and 43=644^3 = 64. Therefore, the numerator becomes:

24â‹…43=16â‹…64=1024{ 2^4 \cdot 4^3 = 16 \cdot 64 = 1024 }

Simplifying the Denominator

To simplify the denominator, we need to calculate the values of 545^4 and 25325^3. We know that 54=6255^4 = 625 and 253=1562525^3 = 15625. Therefore, the denominator becomes:

54â‹…253=625â‹…15625=9765625{ 5^4 \cdot 25^3 = 625 \cdot 15625 = 9765625 }

Putting it All Together

Now that we have simplified the numerator and the denominator, we can put the expression together:

24â‹…4354â‹…253=10249765625{ \frac{2^4 \cdot 4^3}{5^4 \cdot 25^3} = \frac{1024}{9765625} }

Simplifying the Fraction

To simplify the fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 1024 and 9765625 is 1. Therefore, the fraction cannot be simplified further.

Conclusion

In this article, we simplified the expression ${ \left(\frac{2}{5}\right)^4 \cdot \left(\frac{4}{25}\right)^3 }$. We broke down the steps involved in simplifying the expression and provided a clear explanation of the mathematical concepts used. We also reviewed the rules of exponentiation and how to manipulate expressions to simplify them. By following these steps, we were able to simplify the expression to ${ \frac{1024}{9765625} }$. This example demonstrates the importance of understanding exponentiation and simplifying expressions in mathematics.

Frequently Asked Questions

  • What is exponentiation? Exponentiation is a mathematical operation that involves raising a number to a power. For example, ${ \left(\frac{2}{5}\right)^4 }$ means ${ \frac{24}{54} }$.
  • How do I simplify a fraction with exponents? To simplify a fraction with exponents, you need to raise both the numerator and the denominator to the power. Then, simplify the resulting fraction by finding the greatest common divisor (GCD) of the numerator and the denominator.
  • What is the greatest common divisor (GCD)? The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder.

Final Thoughts

Simplifying expressions is an essential skill in mathematics that helps us solve problems efficiently and accurately. By understanding the rules of exponentiation and how to manipulate expressions, we can simplify complex expressions and arrive at the correct solution. In this article, we simplified the expression ${ \left(\frac{2}{5}\right)^4 \cdot \left(\frac{4}{25}\right)^3 }$. We hope that this example has helped you understand the importance of simplifying expressions in mathematics.

Introduction

In our previous article, we simplified the expression \[(25)4⋅(425)3 ]\[ \left(\frac{2}{5}\right)^4 \cdot \left(\frac{4}{25}\right)^3 \ ]. We broke down the steps involved in simplifying the expression and provided a clear explanation of the mathematical concepts used. In this article, we will answer some frequently asked questions related to simplifying expressions with exponents.

Q&A

Q: What is exponentiation?

A: Exponentiation is a mathematical operation that involves raising a number to a power. For example, \[ \left(\frac{2}{5}\right)^4 }$ means ${ \frac{24}{54} }$.

Q: How do I simplify a fraction with exponents?

A: To simplify a fraction with exponents, you need to raise both the numerator and the denominator to the power. Then, simplify the resulting fraction by finding the greatest common divisor (GCD) of the numerator and the denominator.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder.

Q: How do I simplify a fraction with multiple exponents?

A: To simplify a fraction with multiple exponents, you need to apply the rules of exponentiation. For example, ${ \left(\frac{2}{5}\right)^4 \cdot \left(\frac{4}{25}\right)^3 \ ]$ can be simplified by raising both the numerator and the denominator to the power.

Q: What is the difference between a power and an exponent?

A: A power is the result of raising a number to a power, while an exponent is the number that is being raised to a power. For example, in the expression \[(25)4 ]\[ \left(\frac{2}{5}\right)^4 \ ], 4 is the exponent and \[ \frac{2}{5} }$ is the base.

Q: How do I simplify a fraction with negative exponents?

A: To simplify a fraction with negative exponents, you need to apply the rule that ${ a^{-n} = \frac{1}{a^n} }$. For example, \[(25)−4 ]\[ \left(\frac{2}{5}\right)^{-4} \ ] can be simplified to \[5424 ]\[ \frac{5^4}{2^4} \ ].

Q: What is the order of operations when simplifying expressions with exponents?

A: The order of operations when simplifying expressions with exponents is:

  1. Evaluate any expressions inside parentheses.
  2. Evaluate any exponents.
  3. Multiply and divide from left to right.
  4. Add and subtract from left to right.

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions with exponents. We provided clear explanations of the mathematical concepts used and examples to illustrate the concepts. By following these steps, you can simplify complex expressions and arrive at the correct solution.

Final Thoughts

Simplifying expressions is an essential skill in mathematics that helps us solve problems efficiently and accurately. By understanding the rules of exponentiation and how to manipulate expressions, we can simplify complex expressions and arrive at the correct solution. In this article, we hope that you have gained a better understanding of how to simplify expressions with exponents.

Additional Resources

  • Mathway: A online math problem solver that can help you simplify expressions with exponents.
  • Khan Academy: A free online resource that provides video lessons and practice exercises on simplifying expressions with exponents.
  • Math Open Reference: A free online reference book that provides detailed explanations and examples of simplifying expressions with exponents.

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