Simplify The Expression 256 ( 5 X − 2 ) 12 4 \sqrt[4]{256(5x-2)^{12}} 4 256 ( 5 X − 2 ) 12 ​ With Positive Rational Exponents.

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Introduction

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In this article, we will simplify the given expression $\sqrt[4]{256(5x-2)^{12}}$ using positive rational exponents. The expression involves a fourth root and a power of 12, which can be simplified using the properties of exponents and roots.

Understanding the Expression

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The given expression is $\sqrt[4]{256(5x-2)^{12}}$. To simplify this expression, we need to understand the properties of roots and exponents. The fourth root of a number is the number that, when raised to the power of 4, gives the original number. In this case, we have a fourth root of 256 multiplied by $(5x-2)^{12}$.

Simplifying the Expression

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To simplify the expression, we can start by simplifying the fourth root of 256. We know that $256 = 4^4$, so we can rewrite the expression as $\sqrt[4]{4^4(5x-2)^{12}}$. Using the property of roots, we can simplify this expression as $4\sqrt[4]{(5x-2)^{12}}$.

Simplifying the Power of 12

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Next, we can simplify the power of 12. We know that $(5x-2)^{12}$ can be rewritten as $((5x-2)^3)^4$. Using the property of exponents, we can simplify this expression as $(5x-2)^{36}$.

Combining the Simplifications

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Now, we can combine the simplifications we made earlier. We have $4\sqrt[4]{(5x-2)^{12}} = 4\sqrt[4]{(5x-2)^{36}}$. Using the property of roots, we can simplify this expression as $4(5x-2)^9$.

Conclusion

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In this article, we simplified the expression $\sqrt[4]{256(5x-2)^{12}}$ using positive rational exponents. We started by simplifying the fourth root of 256 and then simplified the power of 12. Finally, we combined the simplifications to get the final expression $4(5x-2)^9$.

Properties of Exponents and Roots

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In this section, we will discuss the properties of exponents and roots that we used to simplify the expression.

Property 1: Roots of Powers

If $a$ is a real number and $n$ is a positive integer, then $\sqrt[n]{a^n} = a$.

Property 2: Powers of Roots

If $a$ is a real number and $n$ is a positive integer, then $(\sqrt[n]{a})^m = a^{m/n}$.

Property 3: Exponents of Powers

If $a$ is a real number and $m$ and $n$ are positive integers, then $(a^m)^n = a^{mn}$.

Examples of Simplifying Expressions with Positive Rational Exponents

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In this section, we will provide examples of simplifying expressions with positive rational exponents.

Example 1

Simplify the expression $\sqrt[3]{27x^6}$.

Solution

Using the property of roots, we can simplify the expression as $\sqrt[3]{27x^6} = 3x^2$.

Example 2

Simplify the expression $\sqrt[4]{16y^8}$.

Solution

Using the property of roots, we can simplify the expression as $\sqrt[4]{16y^8} = 2y^2$.

Conclusion

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In this article, we simplified the expression $\sqrt[4]{256(5x-2)^{12}}$ using positive rational exponents. We discussed the properties of exponents and roots that we used to simplify the expression and provided examples of simplifying expressions with positive rational exponents.

Final Answer

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The final answer is $\boxed{4(5x-2)^9}$.

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Introduction

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In our previous article, we simplified the expression $\sqrt[4]{256(5x-2)^{12}}$ using positive rational exponents. In this article, we will answer some frequently asked questions about simplifying expressions with positive rational exponents.

Q&A

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Q: What is a positive rational exponent?

A: A positive rational exponent is a fraction that represents the power to which a number is raised. For example, $2^{3/4}$ is a positive rational exponent.

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

A: To simplify an expression with a positive rational exponent, you can use the properties of exponents and roots. For example, if you have the expression $\sqrt[3]{27x^6}$, you can simplify it as $3x^2$.

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

A: A root is the number that, when raised to a certain power, gives the original number. For example, the fourth root of 256 is 4, because $4^4 = 256$. An exponent, on the other hand, is the power to which a number is raised. For example, $2^3 = 8$.

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

A: To simplify an expression with a negative exponent, you can use the property of exponents that states $a^{-n} = 1/a^n$. For example, if you have the expression $2^{-3}$, you can simplify it as $1/2^3 = 1/8$.

Q: Can I simplify an expression with a fractional exponent?

A: Yes, you can simplify an expression with a fractional exponent. For example, if you have the expression $\sqrt[3]{27x^6}$, you can simplify it as $3x^2$.

Q: How do I simplify an expression with a variable in the exponent?

A: To simplify an expression with a variable in the exponent, you can use the properties of exponents and roots. For example, if you have the expression $\sqrt[3]{27x^6}$, you can simplify it as $3x^2$.

Q: Can I simplify an expression with a negative variable in the exponent?

A: Yes, you can simplify an expression with a negative variable in the exponent. For example, if you have the expression $2^{-3x}$, you can simplify it as $1/2^{3x}$.

Examples of Simplifying Expressions with Positive Rational Exponents

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In this section, we will provide examples of simplifying expressions with positive rational exponents.

Example 1

Simplify the expression $\sqrt[3]{27x^6}$.

Solution

Using the property of roots, we can simplify the expression as $\sqrt[3]{27x^6} = 3x^2$.

Example 2

Simplify the expression $\sqrt[4]{16y^8}$.

Solution

Using the property of roots, we can simplify the expression as $\sqrt[4]{16y^8} = 2y^2$.

Example 3

Simplify the expression $\sqrt[3]{27x^6y^3}$.

Solution

Using the property of roots, we can simplify the expression as $\sqrt[3]{27x^6y^3} = 3x^2y$.

Conclusion

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In this article, we answered some frequently asked questions about simplifying expressions with positive rational exponents. We discussed the properties of exponents and roots that we used to simplify the expressions and provided examples of simplifying expressions with positive rational exponents.

Final Answer

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The final answer is $\boxed{3x^2y}$.