Which Expressions Are Equivalent To The One Below? Check All That Apply. 36 Z 6 Z \frac{36^z}{6^z} 6 Z 3 6 Z ​ A. { (36-6)^x$}$B. 6C. { \frac{6^z \cdot 6 Z}{6 Z}$}$D. ${ 6^x\$} E. { \left(\frac{36}{6}\right)^x$}$F.

Simplifying Exponential Expressions: Which Ones Are Equivalent?

When dealing with exponential expressions, it's essential to understand the properties of exponents and how to simplify them. In this article, we'll explore the given expression $\frac{36^z}{6^z}$ and determine which of the provided options are equivalent to it.

Understanding Exponential Expressions

Before we dive into the given expression, let's review the basics of exponential expressions. An exponential expression is a mathematical expression that represents a quantity raised to a power. For example, $2^3$ represents $2$ raised to the power of $3$. Exponential expressions can be written in the form $a^b$, where $a$ is the base and $b$ is the exponent.

The Given Expression

The given expression is $\frac{36^z}{6^z}$. To simplify this expression, we can use the property of exponents that states $\frac{a^m}{a^n} = a^{m-n}$, where $a$ is the base and $m$ and $n$ are the exponents.

Using this property, we can rewrite the given expression as:

$$\frac{36^z}{6^z} = \frac{(6^2)^z}{6^z} = 6^{2z-z} = 6^z$$

Option A: $(36-6)^x$

Option A is $(36-6)^x$. To determine if this option is equivalent to the given expression, we can simplify it using the property of exponents that states $(a-b)^n = a^n - b^n$, where $a$ and $b$ are the bases and $n$ is the exponent.

Using this property, we can rewrite option A as:

$$(36-6)^x = (30)^x = 6^x \cdot 5^x$$

Since this expression is not equivalent to the given expression, option A is not correct.

Option B: 6

Option B is simply $6$. To determine if this option is equivalent to the given expression, we can rewrite the given expression as:

$$\frac{36^z}{6^z} = 6^z$$

Since option B is not equal to $6^z$, option B is not correct.

Option C: $\frac{6^z \cdot 6^z}{6^z}$

Option C is $\frac{6^z \cdot 6^z}{6^z}$. To determine if this option is equivalent to the given expression, we can simplify it using the property of exponents that states $\frac{a^m \cdot a^n}{a^m} = a^n$, where $a$ is the base and $m$ and $n$ are the exponents.

Using this property, we can rewrite option C as:

$$\frac{6^z \cdot 6^z}{6^z} = 6^z$$

Since option C is equivalent to the given expression, option C is correct.

Option D: $6^x$

Option D is $6^x$. To determine if this option is equivalent to the given expression, we can rewrite the given expression as:

$$\frac{36^z}{6^z} = 6^z$$

Since option D is not equal to $6^z$, option D is not correct.

Option E: $\left(\frac{36}{6}\right)^x$

Option E is $\left(\frac{36}{6}\right)^x$. To determine if this option is equivalent to the given expression, we can simplify it using the property of exponents that states $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$, where $a$ and $b$ are the bases and $n$ is the exponent.

Using this property, we can rewrite option E as:

$$\left(\frac{36}{6}\right)^x = \frac{36^x}{6^x}$$

Since this expression is not equivalent to the given expression, option E is not correct.

Conclusion

In conclusion, the only option that is equivalent to the given expression $\frac{36^z}{6^z}$ is option C: $\frac{6^z \cdot 6^z}{6^z}$. This option can be simplified to $6^z$, which is equivalent to the given expression.

Final Answer

The final answer is:

• Option C: $\frac{6^z \cdot 6^z}{6^z}$
  Simplifying Exponential Expressions: A Q&A Guide

In our previous article, we explored the given expression $\frac{36^z}{6^z}$ and determined which of the provided options are equivalent to it. In this article, we'll answer some frequently asked questions about simplifying exponential expressions.

Q: What is the property of exponents that states $\frac{a^m}{a^n} = a^{m-n}$?

A: This property is known as the quotient rule of exponents. It states that when we divide two exponential expressions with the same base, we can subtract the exponents.

Q: How do I simplify an exponential expression like $\frac{36^z}{6^z}$?

A: To simplify this expression, we can use the quotient rule of exponents. We can rewrite the expression as:

$$\frac{36^z}{6^z} = \frac{(6^2)^z}{6^z} = 6^{2z-z} = 6^z$$

Q: What is the difference between $6^x$ and $6^z$?

A: The difference between $6^x$ and $6^z$ is the value of the exponent. In $6^x$, the exponent is $x$, while in $6^z$, the exponent is $z$. These two expressions are not equivalent, even if they have the same base.

Q: Can I simplify an exponential expression like $(36-6)^x$?

A: Yes, you can simplify this expression using the property of exponents that states $(a-b)^n = a^n - b^n$. However, this expression is not equivalent to the given expression $\frac{36^z}{6^z}$.

Q: What is the property of exponents that states $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$?

A: This property is known as the power rule of exponents. It states that when we raise a fraction to a power, we can raise the numerator and denominator to that power.

Q: How do I simplify an exponential expression like $\left(\frac{36}{6}\right)^x$?

A: To simplify this expression, we can use the power rule of exponents. We can rewrite the expression as:

$$\left(\frac{36}{6}\right)^x = \frac{36^x}{6^x}$$

Q: What is the difference between $\frac{6^z \cdot 6^z}{6^z}$ and $6^z$?

A: The difference between these two expressions is that $\frac{6^z \cdot 6^z}{6^z}$ is equivalent to $6^z$, while $6^z$ is a simplified version of the original expression.

Q: Can I simplify an exponential expression like $\frac{6^z \cdot 6^z}{6^z}$?

A: Yes, you can simplify this expression using the property of exponents that states $\frac{a^m \cdot a^n}{a^m} = a^n$. This expression is equivalent to $6^z$.

Conclusion

In conclusion, simplifying exponential expressions requires a good understanding of the properties of exponents. By using the quotient rule, power rule, and other properties, we can simplify complex expressions and make them easier to work with.

Common Mistakes to Avoid

When simplifying exponential expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not using the correct property of exponents
• Not simplifying the expression correctly
• Not checking the equivalence of the expression

Tips for Simplifying Exponential Expressions

Here are some tips for simplifying exponential expressions:

• Use the quotient rule to simplify expressions with the same base
• Use the power rule to simplify expressions with a fraction as the base
• Check the equivalence of the expression before simplifying it
• Use the correct property of exponents to simplify the expression

Final Answer

The final answer is:

• Use the quotient rule to simplify expressions with the same base
• Use the power rule to simplify expressions with a fraction as the base
• Check the equivalence of the expression before simplifying it
• Use the correct property of exponents to simplify the expression