Simplify: 64 X 17 \sqrt{64 X^{17}} 64 X 17 ​ Assume That The Variable Represents (complete The Assumption As Needed).

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Understanding the Problem

To simplify the given expression, we need to first understand the properties of square roots and exponents. The expression $\sqrt{64 x^{17}}$ involves both a numerical coefficient and a variable raised to a power. We will assume that the variable $x$ represents a non-negative real number, as the square root of a negative number is undefined in the real number system.

Breaking Down the Expression

The expression $\sqrt{64 x^{17}}$ can be broken down into two parts: the numerical coefficient $64$ and the variable $x^{17}$. We can simplify the numerical coefficient by finding its square root, which is $8$. The variable $x^{17}$ can be simplified by using the property of exponents that states $\sqrt{x^n} = x^{n/2}$.

Simplifying the Numerical Coefficient

The numerical coefficient $64$ can be simplified by finding its square root, which is $8$. This is because $8^2 = 64$, so the square root of $64$ is $8$.

Simplifying the Variable

The variable $x^{17}$ can be simplified by using the property of exponents that states $\sqrt{x^n} = x^{n/2}$. In this case, $n = 17$, so we can simplify the variable as follows:

$$\sqrt{x^{17}} = x^{17/2} = x^{8.5}$$

Combining the Simplified Parts

Now that we have simplified the numerical coefficient and the variable, we can combine the two parts to get the final simplified expression:

$$\sqrt{64 x^{17}} = 8x^{8.5}$$

Conclusion

In conclusion, the simplified expression for $\sqrt{64 x^{17}}$ is $8x^{8.5}$. This expression involves both a numerical coefficient and a variable raised to a power. We assumed that the variable $x$ represents a non-negative real number, as the square root of a negative number is undefined in the real number system.

Example Use Case

Suppose we want to simplify the expression $\sqrt{64 x^{17}}$ when $x = 2$. We can plug in the value of $x$ into the simplified expression to get:

$$8(2)^{8.5} = 8(2^8)(2^{0.5}) = 8(256)(\sqrt{2}) = 2048\sqrt{2}$$

This example illustrates how to simplify the expression $\sqrt{64 x^{17}}$ when the variable $x$ is given a specific value.

Properties of Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of $16$ is $4$, because $4 \times 4 = 16$. The square root of a negative number is undefined in the real number system, but it can be represented using complex numbers.

Properties of Exponents

Exponents are a shorthand way of writing repeated multiplication. For example, $x^3$ means $x \times x \times x$. The property of exponents that states $\sqrt{x^n} = x^{n/2}$ is a useful tool for simplifying expressions involving square roots and exponents.

Final Thoughts

In this article, we simplified the expression $\sqrt{64 x^{17}}$ by breaking it down into two parts: the numerical coefficient $64$ and the variable $x^{17}$. We then simplified each part using the properties of square roots and exponents. The final simplified expression is $8x^{8.5}$. This expression involves both a numerical coefficient and a variable raised to a power, and it assumes that the variable $x$ represents a non-negative real number.

Simplifying Expressions Involving Square Roots and Exponents

Simplifying expressions involving square roots and exponents can be a challenging task, but it can be made easier by using the properties of square roots and exponents. By breaking down the expression into two parts and simplifying each part separately, we can arrive at the final simplified expression.

Example of Simplifying an Expression Involving Square Roots and Exponents

Suppose we want to simplify the expression $\sqrt{64 x^{17}}$ when $x = 2$. We can plug in the value of $x$ into the simplified expression to get:

$$8(2)^{8.5} = 8(2^8)(2^{0.5}) = 8(256)(\sqrt{2}) = 2048\sqrt{2}$$

This example illustrates how to simplify the expression $\sqrt{64 x^{17}}$ when the variable $x$ is given a specific value.

Tips for Simplifying Expressions Involving Square Roots and Exponents

1. Break down the expression into two parts: The numerical coefficient and the variable.
2. Simplify each part separately: Use the properties of square roots and exponents to simplify each part.
3. Combine the simplified parts: The final simplified expression is the product of the simplified numerical coefficient and the simplified variable.

By following these tips, we can simplify expressions involving square roots and exponents with ease.

Conclusion

In conclusion, simplifying expressions involving square roots and exponents can be a challenging task, but it can be made easier by using the properties of square roots and exponents. By breaking down the expression into two parts and simplifying each part separately, we can arrive at the final simplified expression. The tips provided in this article can be used to simplify expressions involving square roots and exponents with ease.

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Frequently Asked Questions

Q: What is the simplified expression for $\sqrt{64 x^{17}}$?

A: The simplified expression for $\sqrt{64 x^{17}}$ is $8x^{8.5}$.

Q: What is the assumption made about the variable $x$?

A: We assumed that the variable $x$ represents a non-negative real number, as the square root of a negative number is undefined in the real number system.

Q: How do you simplify the numerical coefficient $64$?

A: The numerical coefficient $64$ can be simplified by finding its square root, which is $8$. This is because $8^2 = 64$, so the square root of $64$ is $8$.

Q: How do you simplify the variable $x^{17}$?

A: The variable $x^{17}$ can be simplified by using the property of exponents that states $\sqrt{x^n} = x^{n/2}$. In this case, $n = 17$, so we can simplify the variable as follows:

$$\sqrt{x^{17}} = x^{17/2} = x^{8.5}$$

Q: What is the final simplified expression for $\sqrt{64 x^{17}}$ when $x = 2$?

A: The final simplified expression for $\sqrt{64 x^{17}}$ when $x = 2$ is:

$$8(2)^{8.5} = 8(2^8)(2^{0.5}) = 8(256)(\sqrt{2}) = 2048\sqrt{2}$$

Q: What are some tips for simplifying expressions involving square roots and exponents?

A: Here are some tips for simplifying expressions involving square roots and exponents:

1. Break down the expression into two parts: The numerical coefficient and the variable.
2. Simplify each part separately: Use the properties of square roots and exponents to simplify each part.
3. Combine the simplified parts: The final simplified expression is the product of the simplified numerical coefficient and the simplified variable.

Q: What is the property of exponents that states $\sqrt{x^n} = x^{n/2}$?

A: This property of exponents states that the square root of a number raised to a power is equal to the number raised to half of that power.

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

A: A square root is a value that, when multiplied by itself, gives the original number. An exponent is a shorthand way of writing repeated multiplication.

Q: Can you provide an example of simplifying an expression involving square roots and exponents?

A: Suppose we want to simplify the expression $\sqrt{64 x^{17}}$ when $x = 2$. We can plug in the value of $x$ into the simplified expression to get:

$$8(2)^{8.5} = 8(2^8)(2^{0.5}) = 8(256)(\sqrt{2}) = 2048\sqrt{2}$$

This example illustrates how to simplify the expression $\sqrt{64 x^{17}}$ when the variable $x$ is given a specific value.

Q: What is the final thought on simplifying expressions involving square roots and exponents?

A: In conclusion, simplifying expressions involving square roots and exponents can be a challenging task, but it can be made easier by using the properties of square roots and exponents. By breaking down the expression into two parts and simplifying each part separately, we can arrive at the final simplified expression. The tips provided in this article can be used to simplify expressions involving square roots and exponents with ease.

Q: What is the next step after simplifying an expression involving square roots and exponents?

A: After simplifying an expression involving square roots and exponents, the next step is to use the simplified expression to solve the problem or equation that it is a part of.

Q: Can you provide a real-world example of simplifying an expression involving square roots and exponents?

A: Suppose we want to simplify the expression $\sqrt{64 x^{17}}$ in the context of a physics problem. We might be given a value for $x$ and asked to find the final simplified expression. In this case, we can use the tips provided in this article to simplify the expression and arrive at the final answer.

Q: What is the importance of simplifying expressions involving square roots and exponents?

A: Simplifying expressions involving square roots and exponents is important because it allows us to solve problems and equations more easily. By simplifying the expression, we can make it easier to work with and understand the underlying math.

Q: Can you provide a summary of the article?

A: In this article, we simplified the expression $\sqrt{64 x^{17}}$ by breaking it down into two parts: the numerical coefficient $64$ and the variable $x^{17}$. We then simplified each part using the properties of square roots and exponents. The final simplified expression is $8x^{8.5}$. This expression involves both a numerical coefficient and a variable raised to a power, and it assumes that the variable $x$ represents a non-negative real number.