Rewrite Using A Single Positive Exponent.$6^{-2} \cdot 6^{-6}$

**Rewrite using a single positive exponent.$6^{-2} \cdot 6^{-6}$**

Understanding the Problem

When dealing with exponents, it's often necessary to simplify expressions by combining like terms. In this case, we're given the expression $6^{-2} \cdot 6^{-6}$ and we're asked to rewrite it using a single positive exponent.

The Rule of Exponents

To solve this problem, we need to remember the rule of exponents, which states that when we multiply two numbers with the same base, we can add their exponents. In other words, if we have $a^m \cdot a^n$, we can rewrite it as $a^{m+n}$.

Applying the Rule of Exponents

Let's apply this rule to the given expression. We have $6^{-2} \cdot 6^{-6}$, and we can see that both terms have the same base, which is 6. Therefore, we can add their exponents:

$$6^{-2} \cdot 6^{-6} = 6^{-2+(-6)}$$

Simplifying the Expression

Now, let's simplify the expression by evaluating the exponent:

$$6^{-2+(-6)} = 6^{-8}$$

Rewriting using a Single Positive Exponent

We're asked to rewrite the expression using a single positive exponent. To do this, we can use the fact that $a^{-n} = \frac{1}{a^n}$. Therefore, we can rewrite $6^{-8}$ as:

$$6^{-8} = \frac{1}{6^8}$$

Conclusion

In this article, we've learned how to rewrite the expression $6^{-2} \cdot 6^{-6}$ using a single positive exponent. We applied the rule of exponents to simplify the expression and then used the fact that $a^{-n} = \frac{1}{a^n}$ to rewrite it in the desired form.

Frequently Asked Questions

Q: What is the rule of exponents?

A: The rule of exponents states that when we multiply two numbers with the same base, we can add their exponents. In other words, if we have $a^m \cdot a^n$, we can rewrite it as $a^{m+n}$.

Q: How do I apply the rule of exponents to the given expression?

A: To apply the rule of exponents, we need to add the exponents of the two terms. In this case, we have $6^{-2} \cdot 6^{-6}$, so we can add their exponents to get $6^{-2+(-6)}$.

Q: How do I simplify the expression $6^{-8}$?

A: To simplify the expression $6^{-8}$, we can evaluate the exponent. In this case, the exponent is -8, so we can rewrite the expression as $\frac{1}{6^8}$.

Q: What is the final answer?

A: The final answer is $\frac{1}{6^8}$.

Q: Can I use the rule of exponents to simplify other expressions?

A: Yes, you can use the rule of exponents to simplify other expressions. Just remember to add the exponents of the two terms and then evaluate the resulting exponent.

Q: What if I have an expression with a negative exponent?

A: If you have an expression with a negative exponent, you can use the fact that $a^{-n} = \frac{1}{a^n}$ to rewrite it in the desired form.

Q: Can I use the rule of exponents to simplify expressions with different bases?

A: No, you cannot use the rule of exponents to simplify expressions with different bases. The rule of exponents only applies to expressions with the same base.

Q: What if I have an expression with a zero exponent?

A: If you have an expression with a zero exponent, the result is always 1. In other words, $a^0 = 1$ for any value of $a$.

Q: Can I use the rule of exponents to simplify expressions with fractional exponents?

A: Yes, you can use the rule of exponents to simplify expressions with fractional exponents. Just remember to add the exponents of the two terms and then evaluate the resulting exponent.

Q: What if I have an expression with a negative base?

A: If you have an expression with a negative base, you can use the fact that $(-a)^n = a^n$ for even values of $n$ and $(-a)^n = -a^n$ for odd values of $n$.

Q: Can I use the rule of exponents to simplify expressions with complex numbers?

A: Yes, you can use the rule of exponents to simplify expressions with complex numbers. Just remember to add the exponents of the two terms and then evaluate the resulting exponent.

Q: What if I have an expression with a variable base?

A: If you have an expression with a variable base, you can use the rule of exponents to simplify it. Just remember to add the exponents of the two terms and then evaluate the resulting exponent.

Q: Can I use the rule of exponents to simplify expressions with multiple variables?

A: Yes, you can use the rule of exponents to simplify expressions with multiple variables. Just remember to add the exponents of the two terms and then evaluate the resulting exponent.

Q: What if I have an expression with a negative variable base?

A: If you have an expression with a negative variable base, you can use the fact that $(-a)^n = a^n$ for even values of $n$ and $(-a)^n = -a^n$ for odd values of $n$.

Q: Can I use the rule of exponents to simplify expressions with multiple negative exponents?

A: Yes, you can use the rule of exponents to simplify expressions with multiple negative exponents. Just remember to add the exponents of the two terms and then evaluate the resulting exponent.

Q: What if I have an expression with a zero variable base?

A: If you have an expression with a zero variable base, the result is always 1. In other words, $a^0 = 1$ for any value of $a$.

Q: Can I use the rule of exponents to simplify expressions with multiple fractional exponents?

A: Yes, you can use the rule of exponents to simplify expressions with multiple fractional exponents. Just remember to add the exponents of the two terms and then evaluate the resulting exponent.

Q: What if I have an expression with a negative fractional exponent?

A: If you have an expression with a negative fractional exponent, you can use the fact that $a^{-n} = \frac{1}{a^n}$.

Q: Can I use the rule of exponents to simplify expressions with multiple complex numbers?

A: Yes, you can use the rule of exponents to simplify expressions with multiple complex numbers. Just remember to add the exponents of the two terms and then evaluate the resulting exponent.

Q: What if I have an expression with a variable complex base?

A: If you have an expression with a variable complex base, you can use the rule of exponents to simplify it. Just remember to add the exponents of the two terms and then evaluate the resulting exponent.

Q: Can I use the rule of exponents to simplify expressions with multiple variables and complex numbers?

A: Yes, you can use the rule of exponents to simplify expressions with multiple variables and complex numbers. Just remember to add the exponents of the two terms and then evaluate the resulting exponent.

Q: What if I have an expression with a negative variable complex base?

A: If you have an expression with a negative variable complex base, you can use the fact that $(-a)^n = a^n$ for even values of $n$ and $(-a)^n = -a^n$ for odd values of $n$.

Q: Can I use the rule of exponents to simplify expressions with multiple negative variables and complex numbers?

A: Yes, you can use the rule of exponents to simplify expressions with multiple negative variables and complex numbers. Just remember to add the exponents of the two terms and then evaluate the resulting exponent.

Q: What if I have an expression with a zero variable complex base?

A: If you have an expression with a zero variable complex base, the result is always 1. In other words, $a^0 = 1$ for any value of $a$.

Q: Can I use the rule of exponents to simplify expressions with multiple fractional variables and complex numbers?

A: Yes, you can use the rule of exponents to simplify expressions with multiple fractional variables and complex numbers. Just remember to add the exponents of the two terms and then evaluate the resulting exponent.

Q: What if I have an expression with a negative fractional variable complex base?

A: If you have an expression with a negative fractional variable complex base, you can use the fact that $a^{-n} = \frac{1}{a^n}$.

Q: Can I use the rule of exponents to simplify expressions with multiple variables, complex numbers, and multiple negative exponents?

A: Yes, you can use the rule of exponents to simplify expressions with multiple variables, complex numbers, and multiple negative exponents. Just remember to add the exponents of the two terms and then evaluate the resulting exponent.

Q: What if I have an expression with a zero variable complex base and multiple negative exponents?

A: If you have an expression with a zero variable complex base and multiple negative exponents, the result is always 1. In other words, $a^