What Is The Value Of The Expression 5 6 5 ⋅ 5 6 5 \sqrt[5]{5^6} \cdot \sqrt[5]{5^6} 5 5 6 ​ ⋅ 5 5 6 ​ ?

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Introduction

In mathematics, expressions involving exponents and roots are common and can be solved using various techniques. One such expression is $\sqrt[5]{5^6} \cdot \sqrt[5]{5^6}$. In this article, we will explore the value of this expression and provide a step-by-step solution.

Understanding Exponents and Roots

Before we dive into the solution, let's briefly review the concepts of exponents and roots.

• Exponents: An exponent is a small number that is raised to the power of a base number. For example, $5^6$ means $5$ raised to the power of $6$. In this case, $5^6 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 15625$.
• Roots: A root is the inverse operation of an exponent. For example, $\sqrt[5]{5^6}$ means finding the fifth root of $5^6$. In this case, $\sqrt[5]{5^6} = \sqrt[5]{15625} = 5$.

Solving the Expression

Now that we have a basic understanding of exponents and roots, let's solve the expression $\sqrt[5]{5^6} \cdot \sqrt[5]{5^6}$.

Step 1: Simplify the Expression

We can start by simplifying the expression using the properties of exponents and roots.

$\sqrt[5]{5^6} \cdot \sqrt[5]{5^6} = \sqrt[5]{5^6} \cdot \sqrt[5]{5^6}$

Using the property of roots, we can rewrite the expression as:

$\sqrt[5]{5^6} \cdot \sqrt[5]{5^6} = \sqrt[5]{5^6 \cdot 5^6}$

Step 2: Combine Exponents

Now, we can combine the exponents using the property of exponents.

$\sqrt[5]{5^6 \cdot 5^6} = \sqrt[5]{5^{6+6}}$

Using the property of exponents, we can simplify the expression as:

$\sqrt[5]{5^{6+6}} = \sqrt[5]{5^{12}}$

Step 3: Evaluate the Root

Finally, we can evaluate the root using the property of roots.

$\sqrt[5]{5^{12}} = \sqrt[5]{(5^6)^2}$

Using the property of roots, we can rewrite the expression as:

$\sqrt[5]{(5^6)^2} = (5^6)^{\frac{2}{5}}$

Step 4: Simplify the Expression

Now, we can simplify the expression using the property of exponents.

$(5^6)^{\frac{2}{5}} = 5^{6 \cdot \frac{2}{5}}$

Using the property of exponents, we can simplify the expression as:

$5^{6 \cdot \frac{2}{5}} = 5^{\frac{12}{5}}$

Step 5: Evaluate the Expression

Finally, we can evaluate the expression using the property of exponents.

$5^{\frac{12}{5}} = 5^2 \cdot 5^{\frac{2}{5}}$

Using the property of exponents, we can simplify the expression as:

$5^2 \cdot 5^{\frac{2}{5}} = 25 \cdot \sqrt[5]{25}$

Step 6: Simplify the Expression

Now, we can simplify the expression using the property of roots.

$25 \cdot \sqrt[5]{25} = 25 \cdot 5$

Using the property of multiplication, we can simplify the expression as:

$25 \cdot 5 = 125$

Conclusion

In this article, we have explored the value of the expression $\sqrt[5]{5^6} \cdot \sqrt[5]{5^6}$. We have used various techniques, including simplifying the expression, combining exponents, evaluating the root, and simplifying the expression. The final value of the expression is $125$.

Final Answer

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Introduction

In our previous article, we explored the value of the expression $\sqrt[5]{5^6} \cdot \sqrt[5]{5^6}$. In this article, we will answer some frequently asked questions related to this expression.

Q: What is the value of $\sqrt[5]{5^6}$?

A: The value of $\sqrt[5]{5^6}$ is $5$. This is because the fifth root of $5^6$ is equal to $5$.

Q: How do you simplify the expression $\sqrt[5]{5^6} \cdot \sqrt[5]{5^6}$?

A: To simplify the expression $\sqrt[5]{5^6} \cdot \sqrt[5]{5^6}$, we can start by combining the exponents using the property of exponents. This gives us $\sqrt[5]{5^6 \cdot 5^6}$. We can then simplify this expression further by combining the exponents, which gives us $\sqrt[5]{5^{12}}$. Finally, we can evaluate the root using the property of roots, which gives us $5^{\frac{12}{5}}$. This can be simplified further to $25 \cdot \sqrt[5]{25}$, which is equal to $125$.

Q: What is the property of exponents that we used to simplify the expression?

A: The property of exponents that we used to simplify the expression is the property of combining exponents. This property states that when we multiply two numbers with the same base, we can add their exponents. In this case, we had $5^6 \cdot 5^6$, which we simplified to $5^{6+6}$, or $5^{12}$.

Q: What is the property of roots that we used to evaluate the root?

A: The property of roots that we used to evaluate the root is the property of roots as powers. This property states that the nth root of a number is equal to the number raised to the power of 1/n. In this case, we had $\sqrt[5]{5^{12}}$, which we evaluated to $(5^6)^{\frac{2}{5}}$.

Q: How do you evaluate the expression $5^{\frac{12}{5}}$?

A: To evaluate the expression $5^{\frac{12}{5}}$, we can use the property of exponents that states that when we raise a number to a power, we can multiply the number by itself as many times as the exponent says. In this case, we had $5^{\frac{12}{5}}$, which we evaluated to $5^2 \cdot 5^{\frac{2}{5}}$. We can then simplify this expression further by multiplying the numbers, which gives us $25 \cdot \sqrt[5]{25}$.

Q: What is the final value of the expression $\sqrt[5]{5^6} \cdot \sqrt[5]{5^6}$?

A: The final value of the expression $\sqrt[5]{5^6} \cdot \sqrt[5]{5^6}$ is $125$.

Conclusion

In this article, we have answered some frequently asked questions related to the expression $\sqrt[5]{5^6} \cdot \sqrt[5]{5^6}$. We have used various techniques, including simplifying the expression, combining exponents, evaluating the root, and simplifying the expression. The final value of the expression is $125$.

Final Answer

The final answer is $\boxed{125}$.