Simplify The Expression:$\left(\frac{3 X^{10} Y^{11}}{x^{10} Y}\right)^0$

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Introduction

When dealing with exponents, it's essential to understand the rules and properties that govern their behavior. In this article, we will focus on simplifying the given expression $\left(\frac{3 x^{10} y^{11}}{x^{10} y}\right)^0$. We will break down the expression, apply the rules of exponents, and arrive at the simplified form.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $x^3$ can be read as "x to the power of 3" and is equivalent to $x \times x \times x$. When dealing with exponents, it's crucial to understand the rules that govern their behavior.

Simplifying the Expression

To simplify the given expression, we need to apply the rules of exponents. The expression $\left(\frac{3 x^{10} y^{11}}{x^{10} y}\right)^0$ can be broken down as follows:

$\left(\frac{3 x^{10} y^{11}}{x^{10} y}\right)^0$

Applying the Zero Exponent Rule

The zero exponent rule states that any non-zero number raised to the power of 0 is equal to 1. This rule can be expressed as:

$a^0 = 1$, where $a$ is a non-zero number.

Simplifying the Expression Using the Zero Exponent Rule

Using the zero exponent rule, we can simplify the expression as follows:

$\left(\frac{3 x^{10} y^{11}}{x^{10} y}\right)^0 = \frac{3 x^{10} y^{11}}{x^{10} y}^0$

Canceling Out the Common Factors

We can simplify the expression further by canceling out the common factors in the numerator and denominator.

$\frac{3 x^{10} y^{11}}{x^{10} y}^0 = \frac{3 x^{10} y^{11}}{x^{10} y} \times \frac{1}{1}$

Applying the Rule of Exponents

The rule of exponents states that when dividing two powers with the same base, we subtract the exponents. This rule can be expressed as:

$\frac{a^m}{a^n} = a^{m-n}$, where $a$ is the base and $m$ and $n$ are the exponents.

Simplifying the Expression Using the Rule of Exponents

Using the rule of exponents, we can simplify the expression as follows:

$\frac{3 x^{10} y^{11}}{x^{10} y} = 3 x^{10-10} y^{11-1}$

Evaluating the Expression

We can evaluate the expression further by simplifying the exponents.

$3 x^{10-10} y^{11-1} = 3 x^0 y^{10}$

Simplifying the Expression

We can simplify the expression further by applying the rule that any non-zero number raised to the power of 0 is equal to 1.

$3 x^0 y^{10} = 3 \times 1 \times y^{10}$

Evaluating the Expression

We can evaluate the expression further by multiplying the numbers.

$3 \times 1 \times y^{10} = 3 y^{10}$

Conclusion

In conclusion, we have simplified the expression $\left(\frac{3 x^{10} y^{11}}{x^{10} y}\right)^0$ using the rules of exponents. We applied the zero exponent rule, canceled out the common factors, and used the rule of exponents to simplify the expression. The final simplified form of the expression is $3 y^{10}$.

Final Answer

The final answer is $\boxed{3 y^{10}}$.

Frequently Asked Questions

Q: What is the zero exponent rule?

A: The zero exponent rule states that any non-zero number raised to the power of 0 is equal to 1.

Q: How do you simplify an expression with a zero exponent?

A: To simplify an expression with a zero exponent, you can apply the zero exponent rule, which states that any non-zero number raised to the power of 0 is equal to 1.

Q: What is the rule of exponents?

A: The rule of exponents states that when dividing two powers with the same base, you subtract the exponents.

Q: How do you simplify an expression using the rule of exponents?

A: To simplify an expression using the rule of exponents, you can divide the exponents of the numerator and denominator and subtract the results.

References

• [1] "Exponents and Exponential Functions" by Math Open Reference
• [2] "Rules of Exponents" by Mathway
• [3] "Simplifying Expressions with Exponents" by Khan Academy
  

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Introduction

In our previous article, we simplified the expression $\left(\frac{3 x^{10} y^{11}}{x^{10} y}\right)^0$ using the rules of exponents. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q&A

Q: What is the zero exponent rule?

A: The zero exponent rule states that any non-zero number raised to the power of 0 is equal to 1. This rule can be expressed as:

$a^0 = 1$, where $a$ is a non-zero number.

Q: How do you simplify an expression with a zero exponent?

A: To simplify an expression with a zero exponent, you can apply the zero exponent rule, which states that any non-zero number raised to the power of 0 is equal to 1.

Q: What is the rule of exponents?

A: The rule of exponents states that when dividing two powers with the same base, you subtract the exponents. This rule can be expressed as:

$\frac{a^m}{a^n} = a^{m-n}$, where $a$ is the base and $m$ and $n$ are the exponents.

Q: How do you simplify an expression using the rule of exponents?

A: To simplify an expression using the rule of exponents, you can divide the exponents of the numerator and denominator and subtract the results.

Q: What is the difference between the zero exponent rule and the rule of exponents?

A: The zero exponent rule states that any non-zero number raised to the power of 0 is equal to 1, while the rule of exponents states that when dividing two powers with the same base, you subtract the exponents.

Q: Can you provide an example of how to simplify an expression using the zero exponent rule?

A: Yes, here's an example:

$\left(\frac{2 x^3 y^4}{x^3 y}\right)^0 = \frac{2 x^3 y^4}{x^3 y} \times \frac{1}{1}$

Using the zero exponent rule, we can simplify the expression as follows:

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

Evaluating the expression, we get:

$2 x^3 y^4 \times 1 = 2 x^3 y^4$

Q: Can you provide an example of how to simplify an expression using the rule of exponents?

A: Yes, here's an example:

$\frac{x^5 y^3}{x^2 y^2} = x^{5-2} y^{3-2}$

Using the rule of exponents, we can simplify the expression as follows:

$x^{5-2} y^{3-2} = x^3 y^1$

Evaluating the expression, we get:

$x^3 y^1 = x^3 y$

Conclusion

In conclusion, we have answered some frequently asked questions related to the simplification of the expression $\left(\frac{3 x^{10} y^{11}}{x^{10} y}\right)^0$. We have discussed the zero exponent rule, the rule of exponents, and provided examples of how to simplify expressions using these rules.

Final Answer

The final answer is $\boxed{3 y^{10}}$.

Frequently Asked Questions

Q: What is the zero exponent rule?

A: The zero exponent rule states that any non-zero number raised to the power of 0 is equal to 1.

Q: How do you simplify an expression with a zero exponent?

A: To simplify an expression with a zero exponent, you can apply the zero exponent rule, which states that any non-zero number raised to the power of 0 is equal to 1.

Q: What is the rule of exponents?

A: The rule of exponents states that when dividing two powers with the same base, you subtract the exponents.

Q: How do you simplify an expression using the rule of exponents?

A: To simplify an expression using the rule of exponents, you can divide the exponents of the numerator and denominator and subtract the results.

References

• [1] "Exponents and Exponential Functions" by Math Open Reference
• [2] "Rules of Exponents" by Mathway
• [3] "Simplifying Expressions with Exponents" by Khan Academy