$\left(\frac{2}{5}\right)^4 \cdot \left(\frac{4}{25}\right)^3$Write In Simplest Form.

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Introduction

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. In this article, we will focus on simplifying the expression (25)4⋅(425)3\left(\frac{2}{5}\right)^4 \cdot \left(\frac{4}{25}\right)^3. We will break down the problem into manageable steps, using the properties of exponents to simplify the expression.

Understanding Exponents

Exponents are a shorthand way of writing repeated multiplication. For example, aba^b can be read as "a to the power of b" and is equivalent to a⋅a⋅a⋅…⋅aa \cdot a \cdot a \cdot \ldots \cdot a (b times). In the expression (25)4\left(\frac{2}{5}\right)^4, the exponent 4 indicates that the fraction 25\frac{2}{5} should be multiplied by itself 4 times.

Simplifying the Expression

To simplify the expression (25)4⋅(425)3\left(\frac{2}{5}\right)^4 \cdot \left(\frac{4}{25}\right)^3, we can start by simplifying each fraction separately.

Simplifying the First Fraction

The first fraction is 25\frac{2}{5}. This fraction cannot be simplified further, as the numerator and denominator have no common factors.

Simplifying the Second Fraction

The second fraction is 425\frac{4}{25}. We can simplify this fraction by dividing both the numerator and denominator by their greatest common factor, which is 1. However, we can simplify it further by dividing both the numerator and denominator by 1 and then by 4. This gives us 16.25\frac{1}{6.25}, but we can simplify it further by multiplying the numerator and denominator by 2. This gives us 212.5\frac{2}{12.5}, but we can simplify it further by dividing both the numerator and denominator by 2.5. This gives us 425\frac{4}{25}.

Simplifying the Expression Using Exponent Rules

Now that we have simplified each fraction, we can use the exponent rules to simplify the expression. The exponent rule for multiplication states that when multiplying two numbers with the same base, we can add their exponents. In this case, the base is 25\frac{2}{5}, so we can add the exponents 4 and 3.

(25)4⋅(425)3=(25)4+3=(25)7\left(\frac{2}{5}\right)^4 \cdot \left(\frac{4}{25}\right)^3 = \left(\frac{2}{5}\right)^{4+3} = \left(\frac{2}{5}\right)^7

Simplifying the Expression Further

Now that we have simplified the expression using the exponent rule, we can simplify it further by evaluating the exponent. To do this, we need to multiply the fraction 25\frac{2}{5} by itself 7 times.

(25)7=25⋅25⋅25⋅25⋅25⋅25⋅25\left(\frac{2}{5}\right)^7 = \frac{2}{5} \cdot \frac{2}{5} \cdot \frac{2}{5} \cdot \frac{2}{5} \cdot \frac{2}{5} \cdot \frac{2}{5} \cdot \frac{2}{5}

Evaluating the Expression

To evaluate the expression, we can multiply the numerators and denominators separately.

25⋅25⋅25⋅25⋅25⋅25⋅25=2⋅2⋅2⋅2⋅2⋅2⋅25⋅5⋅5⋅5⋅5⋅5⋅5\frac{2}{5} \cdot \frac{2}{5} \cdot \frac{2}{5} \cdot \frac{2}{5} \cdot \frac{2}{5} \cdot \frac{2}{5} \cdot \frac{2}{5} = \frac{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2}{5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5}

Simplifying the Numerator and Denominator

Now that we have multiplied the numerators and denominators, we can simplify them by combining like terms.

2⋅2⋅2⋅2⋅2⋅2⋅25⋅5⋅5⋅5⋅5⋅5⋅5=12878125\frac{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2}{5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5} = \frac{128}{78125}

Final Answer

The final answer is 12878125\boxed{\frac{128}{78125}}.

Conclusion

In this article, we simplified the expression (25)4⋅(425)3\left(\frac{2}{5}\right)^4 \cdot \left(\frac{4}{25}\right)^3 using the properties of exponents. We started by simplifying each fraction separately, and then used the exponent rule for multiplication to simplify the expression. Finally, we evaluated the expression by multiplying the numerators and denominators, and simplified the result by combining like terms. The final answer is 12878125\boxed{\frac{128}{78125}}.

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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 using the properties of exponents. In this article, we will answer some common questions related to simplifying exponents.

Q&A

Q: What is the rule for multiplying exponents with the same base?

A: When multiplying two numbers with the same base, we can add their exponents. For example, (25)4⋅(25)3=(25)4+3=(25)7\left(\frac{2}{5}\right)^4 \cdot \left(\frac{2}{5}\right)^3 = \left(\frac{2}{5}\right)^{4+3} = \left(\frac{2}{5}\right)^7.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you can start by simplifying each fraction separately. Then, use the exponent rule for multiplication to simplify the expression. Finally, evaluate the expression by multiplying the numerators and denominators, and simplify the result by combining like terms.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the base should be multiplied by itself a certain number of times. For example, (25)4\left(\frac{2}{5}\right)^4 means that the fraction 25\frac{2}{5} should be multiplied by itself 4 times. A negative exponent indicates that the base should be divided by itself a certain number of times. For example, (25)−4\left(\frac{2}{5}\right)^{-4} means that the fraction 25\frac{2}{5} should be divided by itself 4 times.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you can use the rule that a−n=1ana^{-n} = \frac{1}{a^n}. For example, (25)−4=1(25)4\left(\frac{2}{5}\right)^{-4} = \frac{1}{\left(\frac{2}{5}\right)^4}.

Q: What is the rule for dividing exponents with the same base?

A: When dividing two numbers with the same base, we can subtract their exponents. For example, (25)4÷(25)3=(25)4−3=(25)1\left(\frac{2}{5}\right)^4 \div \left(\frac{2}{5}\right)^3 = \left(\frac{2}{5}\right)^{4-3} = \left(\frac{2}{5}\right)^1.

Q: How do I simplify an expression with exponents and fractions?

A: To simplify an expression with exponents and fractions, you can start by simplifying each fraction separately. Then, use the exponent rule for multiplication and division to simplify the expression. Finally, evaluate the expression by multiplying the numerators and denominators, and simplify the result by combining like terms.

Conclusion

In this article, we answered some common questions related to simplifying exponents. We covered topics such as multiplying and dividing exponents with the same base, simplifying expressions with negative exponents, and simplifying expressions with exponents and fractions. By following these rules and guidelines, you can simplify complex expressions with exponents and fractions.

Additional Resources

Final Answer

The final answer is 12878125\boxed{\frac{128}{78125}}.