Simplify The Expression $\sqrt{w^8}$. Assume That The Variable $w$ Represents A Positive Real Number.

$$

Introduction

In this article, we will simplify the expression $\sqrt{w^8}$, assuming that the variable $w$ represents a positive real number. This problem involves the use of exponent rules and properties of square roots. We will break down the expression into simpler terms and provide a step-by-step solution to simplify it.

Understanding Exponent Rules

Before we start simplifying the expression, let's review some exponent rules that will be useful in this problem. The exponent rule states that for any non-zero number $a$ and integers $m$ and $n$, we have:

$a^{m+n} = a^m \cdot a^n$

This rule allows us to combine exponents when we have a product of powers with the same base.

Simplifying the Expression

Now, let's simplify the expression $\sqrt{w^8}$. We can start by rewriting the expression as:

$\sqrt{w^8} = (w^8)^{\frac{1}{2}}$

Using the exponent rule, we can rewrite the expression as:

$(w^8)^{\frac{1}{2}} = w^{8 \cdot \frac{1}{2}}$

Simplifying the exponent, we get:

$w^{8 \cdot \frac{1}{2}} = w^4$

Therefore, the simplified expression is $w^4$.

Properties of Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. In other words, if $x = \sqrt{y}$, then $x^2 = y$. This property is useful in simplifying expressions involving square roots.

Example

Let's consider an example to illustrate the simplification process. Suppose we want to simplify the expression $\sqrt{16}$. Using the exponent rule, we can rewrite the expression as:

$\sqrt{16} = (16)^{\frac{1}{2}}$

Simplifying the exponent, we get:

$(16)^{\frac{1}{2}} = 4$

Therefore, the simplified expression is $4$.

Conclusion

In this article, we simplified the expression $\sqrt{w^8}$, assuming that the variable $w$ represents a positive real number. We used exponent rules and properties of square roots to break down the expression into simpler terms. The simplified expression is $w^4$. We also reviewed some exponent rules and properties of square roots to provide a better understanding of the simplification process.

Exercises

1. Simplify the expression $\sqrt{x^6}$.
2. Simplify the expression $\sqrt{y^9}$.
3. Simplify the expression $\sqrt{z^{12}}$.

Answer Key

1. $x^3$
2. $y^4.5$
3. $z^6$

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak

Note

$$ $$

Introduction

In our previous article, we simplified the expression $\sqrt{w^8}$, assuming that the variable $w$ represents a positive real number. We used exponent rules and properties of square roots to break down the expression into simpler terms. In this article, we will provide a Q&A section to help readers understand the simplification process and address any questions they may have.

Q&A

Q: What is the simplified expression for $\sqrt{w^8}$?

A: The simplified expression is $w^4$.

Q: Why did we use the exponent rule to simplify the expression?

A: We used the exponent rule to simplify the expression because it allows us to combine exponents when we have a product of powers with the same base.

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

A: We used the property of square roots that states that if $x = \sqrt{y}$, then $x^2 = y$.

Q: Can we simplify the expression $\sqrt{x^6}$ using the same method?

A: Yes, we can simplify the expression $\sqrt{x^6}$ using the same method. The simplified expression is $x^3$.

Q: What is the difference between $\sqrt{w^8}$ and $\sqrt{w^4}$?

A: The difference between $\sqrt{w^8}$ and $\sqrt{w^4}$ is that $\sqrt{w^8}$ is equal to $w^4$, while $\sqrt{w^4}$ is equal to $w^2$.

Q: Can we simplify the expression $\sqrt{y^9}$ using the same method?

A: Yes, we can simplify the expression $\sqrt{y^9}$ using the same method. The simplified expression is $y^4.5$.

Q: What is the property of exponents that we used to simplify the expression $\sqrt{z^{12}}$?

A: We used the property of exponents that states that for any non-zero number $a$ and integers $m$ and $n$, we have $a^{m+n} = a^m \cdot a^n$.

Q: Can we simplify the expression $\sqrt{z^{12}}$ using the same method?

A: Yes, we can simplify the expression $\sqrt{z^{12}}$ using the same method. The simplified expression is $z^6$.

Conclusion

In this article, we provided a Q&A section to help readers understand the simplification process and address any questions they may have. We reviewed the exponent rules and properties of square roots that we used to simplify the expression $\sqrt{w^8}$. We also provided examples to illustrate the simplification process and answered questions about the difference between $\sqrt{w^8}$ and $\sqrt{w^4}$.

Exercises

1. Simplify the expression $\sqrt{x^6}$.
2. Simplify the expression $\sqrt{y^9}$.
3. Simplify the expression $\sqrt{z^{12}}$.

Answer Key

1. $x^3$
2. $y^4.5$
3. $z^6$

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak

Note

The variable $w$ represents a positive real number.