Simplify The Expression:${ \left(5 X^2 Y^{\frac{-3}{2}} Z {\frac{1}{4}}\right) 2 }$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve complex problems and understand the underlying concepts. When dealing with exponents and variables, simplifying expressions can be a challenging task. In this article, we will focus on simplifying the given expression $\left(5 x^2 y^{\frac{-3}{2}} z^{\frac{1}{4}}\right)^2$ using the properties of exponents and variables.

Understanding Exponents and Variables

Before we dive into simplifying the expression, let's review the basics of exponents and variables. Exponents are a shorthand way of representing repeated multiplication. For example, $x^2$ means $x$ multiplied by itself, which is equal to $x \times x$. Variables, on the other hand, are letters or symbols that represent unknown values. In the given expression, $x$, $y$, and $z$ are variables.

Simplifying the Expression

To simplify the expression $\left(5 x^2 y^{\frac{-3}{2}} z^{\frac{1}{4}}\right)^2$, we need to apply the properties of exponents. When an exponent is raised to another exponent, we multiply the exponents. In this case, the exponent is 2, so we multiply the exponents of $x$, $y$, and $z$ by 2.

Simplifying the Exponents

Let's simplify the exponents of $x$, $y$, and $z$ separately.

Simplifying the Exponent of $x$

The exponent of $x$ is $2$. When we raise $x^2$ to the power of 2, we multiply the exponent by 2, resulting in $x^{2 \times 2} = x^4$.

Simplifying the Exponent of $y$

The exponent of $y$ is $\frac{-3}{2}$. When we raise $y^{\frac{-3}{2}}$ to the power of 2, we multiply the exponent by 2, resulting in $y^{\frac{-3}{2} \times 2} = y^{-3}$.

Simplifying the Exponent of $z$

The exponent of $z$ is $\frac{1}{4}$. When we raise $z^{\frac{1}{4}}$ to the power of 2, we multiply the exponent by 2, resulting in $z^{\frac{1}{4} \times 2} = z^{\frac{1}{2}}$.

Combining the Simplified Exponents

Now that we have simplified the exponents of $x$, $y$, and $z$, we can combine them to get the final simplified expression.

$\left(5 x^2 y^{\frac{-3}{2}} z^{\frac{1}{4}}\right)^2 = 5^2 \times x^{2 \times 2} \times y^{\frac{-3}{2} \times 2} \times z^{\frac{1}{4} \times 2}$

$= 25 \times x^4 \times y^{-3} \times z^{\frac{1}{2}}$

$= \frac{25 x^4 z^{\frac{1}{2}}}{y^3}$

Conclusion

In this article, we simplified the expression $\left(5 x^2 y^{\frac{-3}{2}} z^{\frac{1}{4}}\right)^2$ using the properties of exponents and variables. We reviewed the basics of exponents and variables, simplified the exponents of $x$, $y$, and $z$ separately, and combined them to get the final simplified expression. The simplified expression is $\frac{25 x^4 z^{\frac{1}{2}}}{y^3}$.

Frequently Asked Questions

• What is the property of exponents that we used to simplify the expression? The property of exponents that we used is the power of a power rule, which states that when an exponent is raised to another exponent, we multiply the exponents.
• How do we simplify the exponents of $x$, $y$, and $z$? We simplify the exponents of $x$, $y$, and $z$ by multiplying the exponents by 2.
• What is the final simplified expression? The final simplified expression is $\frac{25 x^4 z^{\frac{1}{2}}}{y^3}$.

Further Reading

If you want to learn more about simplifying expressions and exponents, here are some recommended resources:

• Khan Academy: Exponents and Variables
• Mathway: Simplifying Expressions
• Wolfram Alpha: Exponents and Variables

References

• [1] Khan Academy. (n.d.). Exponents and Variables. Retrieved from https://www.khanacademy.org/math/algebra/x2f-exponents-and-variables
• [2] Mathway. (n.d.). Simplifying Expressions. Retrieved from https://www.mathway.com/subjects/simplifying-expressions
• [3] Wolfram Alpha. (n.d.). Exponents and Variables. Retrieved from https://www.wolframalpha.com/input/?i=exponents+and+variables
  

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Introduction

In our previous article, we simplified the expression $\left(5 x^2 y^{\frac{-3}{2}} z^{\frac{1}{4}}\right)^2$ using the properties of exponents and variables. In this article, we will answer some frequently asked questions related to simplifying expressions and exponents.

Q&A

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

A: The property of exponents that we used is the power of a power rule, which states that when an exponent is raised to another exponent, we multiply the exponents.

Q: How do we simplify the exponents of $x$, $y$, and $z$?

A: We simplify the exponents of $x$, $y$, and $z$ by multiplying the exponents by 2.

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{25 x^4 z^{\frac{1}{2}}}{y^3}$.

Q: Can we simplify expressions with negative exponents?

A: Yes, we can simplify expressions with negative exponents. When we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base.

Q: How do we simplify expressions with fractional exponents?

A: We can simplify expressions with fractional exponents by using the properties of exponents. For example, $x^{\frac{1}{2}}$ can be rewritten as $\sqrt{x}$.

Q: Can we simplify expressions with multiple variables?

A: Yes, we can simplify expressions with multiple variables. We can use the properties of exponents to simplify the expression, and then combine the variables.

Q: How do we simplify expressions with coefficients?

A: We can simplify expressions with coefficients by using the properties of exponents. For example, $5^2 \times x^2$ can be rewritten as $25x^2$.

Q: Can we simplify expressions with radicals?

A: Yes, we can simplify expressions with radicals. We can use the properties of radicals to simplify the expression, and then combine the variables.

Examples

Example 1: Simplify the expression $\left(2 x^3 y^{\frac{1}{2}} z^{\frac{2}{3}}\right)^2$

A: To simplify the expression, we multiply the exponents of $x$, $y$, and $z$ by 2.

$\left(2 x^3 y^{\frac{1}{2}} z^{\frac{2}{3}}\right)^2 = 2^2 \times x^{3 \times 2} \times y^{\frac{1}{2} \times 2} \times z^{\frac{2}{3} \times 2}$

$= 4 \times x^6 \times y^1 \times z^{\frac{4}{3}}$

$= \frac{4 x^6 z^{\frac{4}{3}}}{y^0}$

$= \frac{4 x^6 z^{\frac{4}{3}}}{1}$

$= 4 x^6 z^{\frac{4}{3}}$

Example 2: Simplify the expression $\left(3 x^2 y^{-2} z^{\frac{1}{2}}\right)^3$

A: To simplify the expression, we multiply the exponents of $x$, $y$, and $z$ by 3.

$\left(3 x^2 y^{-2} z^{\frac{1}{2}}\right)^3 = 3^3 \times x^{2 \times 3} \times y^{-2 \times 3} \times z^{\frac{1}{2} \times 3}$

$= 27 \times x^6 \times y^{-6} \times z^{\frac{3}{2}}$

$= \frac{27 x^6 z^{\frac{3}{2}}}{y^6}$

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions and exponents. We also provided examples of how to simplify expressions with multiple variables, coefficients, and radicals. By following the properties of exponents and variables, we can simplify complex expressions and understand the underlying concepts.

Further Reading

If you want to learn more about simplifying expressions and exponents, here are some recommended resources:

• Khan Academy: Exponents and Variables
• Mathway: Simplifying Expressions
• Wolfram Alpha: Exponents and Variables

References

• [1] Khan Academy. (n.d.). Exponents and Variables. Retrieved from https://www.khanacademy.org/math/algebra/x2f-exponents-and-variables
• [2] Mathway. (n.d.). Simplifying Expressions. Retrieved from https://www.mathway.com/subjects/simplifying-expressions
• [3] Wolfram Alpha. (n.d.). Exponents and Variables. Retrieved from https://www.wolframalpha.com/input/?i=exponents+and+variables