Which Of The Following Is Equal To The Expression Below? ( 6 − 8 ) 4 \left(6^{-8}\right)^4 ( 6 − 8 ) 4 A. 1 4 2 X \frac{1}{4^{2x}} 4 2 X 1 ​ B. -32 C. 1 8 5 \frac{1}{8^5} 8 5 1 ​ D. -6^4

Introduction

Exponential expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will focus on simplifying the expression $\left(6^{-8}\right)^4$ and compare it with the given options to determine which one is equal to it.

Understanding Exponents

Before we dive into simplifying the expression, let's review the basics of exponents. An exponent is a small number that is placed above and to the right of a number or expression, indicating how many times the number or expression should be multiplied by itself. For example, $a^b$ means $a$ multiplied by itself $b$ times.

Simplifying the Expression

Now, let's simplify the expression $\left(6^{-8}\right)^4$. To do this, we need to apply the power rule of exponents, which states that $(a^m)^n = a^{m \cdot n}$. In this case, we have $(6^{-8})^4$, so we can rewrite it as $6^{-8 \cdot 4}$.

(6^{-8})^4 = 6^{-8 \cdot 4} = 6^{-32}

Now, we can simplify the expression further by applying the rule for negative exponents, which states that $a^{-n} = \frac{1}{a^n}$. In this case, we have $6^{-32}$, so we can rewrite it as $\frac{1}{6^{32}}$.

6^{-32} = \frac{1}{6^{32}}

Comparing with the Options

Now that we have simplified the expression, let's compare it with the given options to determine which one is equal to it.

Option A: $\frac{1}{4^{2x}}$

To compare this option with our simplified expression, we need to rewrite it in terms of base 6. We can do this by applying the rule for exponents, which states that $(a^m)^n = a^{m \cdot n}$. In this case, we have $4^{2x}$, so we can rewrite it as $(2^2)^{2x} = 2^{4x}$.

4^{2x} = (2^2)^{2x} = 2^{4x}

Now, we can rewrite the option in terms of base 6 by applying the rule for exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$. In this case, we have $\frac{1}{4^{2x}}$, so we can rewrite it as $\frac{1}{(2^2)^{2x}} = \frac{1}{2^{4x}}$.

\frac{1}{4^{2x}} = \frac{1}{(2^2)^{2x}} = \frac{1}{2^{4x}}

Now, we can compare this option with our simplified expression. We can rewrite our simplified expression in terms of base 2 by applying the rule for exponents, which states that $a^m \cdot a^n = a^{m+n}$. In this case, we have $6^{32}$, so we can rewrite it as $2^{64} \cdot 3^{32}$.

6^{32} = 2^{64} \cdot 3^{32}

Now, we can rewrite our simplified expression in terms of base 2 by applying the rule for exponents, which states that $\frac{1}{a^m} = a^{-m}$. In this case, we have $\frac{1}{6^{32}}$, so we can rewrite it as $\frac{1}{2^{64} \cdot 3^{32}}$.

\frac{1}{6^{32}} = \frac{1}{2^{64} \cdot 3^{32}}

Now, we can compare this option with our simplified expression. We can see that the two expressions are not equal, so option A is not correct.

Option B: -32

This option is a constant value, and it is not equal to our simplified expression, so it is not correct.

Option C: $\frac{1}{8^5}$

To compare this option with our simplified expression, we need to rewrite it in terms of base 6. We can do this by applying the rule for exponents, which states that $(a^m)^n = a^{m \cdot n}$. In this case, we have $8^5$, so we can rewrite it as $(2^3)^5 = 2^{15}$.

8^5 = (2^3)^5 = 2^{15}

Now, we can rewrite the option in terms of base 6 by applying the rule for exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$. In this case, we have $\frac{1}{8^5}$, so we can rewrite it as $\frac{1}{(2^3)^5} = \frac{1}{2^{15}}$.

\frac{1}{8^5} = \frac{1}{(2^3)^5} = \frac{1}{2^{15}}

Now, we can compare this option with our simplified expression. We can rewrite our simplified expression in terms of base 2 by applying the rule for exponents, which states that $a^m \cdot a^n = a^{m+n}$. In this case, we have $6^{32}$, so we can rewrite it as $2^{64} \cdot 3^{32}$.

6^{32} = 2^{64} \cdot 3^{32}

Now, we can rewrite our simplified expression in terms of base 2 by applying the rule for exponents, which states that $\frac{1}{a^m} = a^{-m}$. In this case, we have $\frac{1}{6^{32}}$, so we can rewrite it as $\frac{1}{2^{64} \cdot 3^{32}}$.

\frac{1}{6^{32}} = \frac{1}{2^{64} \cdot 3^{32}}

Now, we can compare this option with our simplified expression. We can see that the two expressions are not equal, so option C is not correct.

Option D: -6^4

This option is a constant value, and it is not equal to our simplified expression, so it is not correct.

Conclusion

In conclusion, we have simplified the expression $\left(6^{-8}\right)^4$ and compared it with the given options to determine which one is equal to it. We found that none of the options are correct, and the expression $\left(6^{-8}\right)^4$ is equal to $\frac{1}{6^{32}}$.

Final Answer

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Q: What is the rule for simplifying exponential expressions?

A: The rule for simplifying exponential expressions is to apply the power rule of exponents, which states that $(a^m)^n = a^{m \cdot n}$. This rule allows us to simplify expressions by combining the exponents.

Q: How do I apply the power rule of exponents?

A: To apply the power rule of exponents, you need to multiply the exponents together. For example, if you have $(a^m)^n$, you can rewrite it as $a^{m \cdot n}$.

Q: What is the rule for negative exponents?

A: The rule for negative exponents is that $a^{-n} = \frac{1}{a^n}$. This rule allows us to rewrite negative exponents as fractions.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you need to rewrite it as a fraction. For example, if you have $a^{-n}$, you can rewrite it as $\frac{1}{a^n}$.

Q: What is the rule for multiplying exponential expressions?

A: The rule for multiplying exponential expressions is that $a^m \cdot a^n = a^{m+n}$. This rule allows us to simplify expressions by combining the exponents.

Q: How do I multiply exponential expressions?

A: To multiply exponential expressions, you need to add the exponents together. For example, if you have $a^m \cdot a^n$, you can rewrite it as $a^{m+n}$.

Q: What is the rule for dividing exponential expressions?

A: The rule for dividing exponential expressions is that $\frac{a^m}{a^n} = a^{m-n}$. This rule allows us to simplify expressions by combining the exponents.

Q: How do I divide exponential expressions?

A: To divide exponential expressions, you need to subtract the exponents. For example, if you have $\frac{a^m}{a^n}$, you can rewrite it as $a^{m-n}$.

Q: Can I simplify an expression with a variable exponent?

A: Yes, you can simplify an expression with a variable exponent. To do this, you need to apply the rules for exponents, such as the power rule and the rule for negative exponents.

Q: How do I simplify an expression with a variable exponent?

A: To simplify an expression with a variable exponent, you need to apply the rules for exponents. For example, if you have $a^{2x}$, you can rewrite it as $(a^2)^x$.

Q: What is the rule for simplifying exponential expressions with fractions?

A: The rule for simplifying exponential expressions with fractions is to apply the rules for exponents, such as the power rule and the rule for negative exponents.

Q: How do I simplify an exponential expression with a fraction?

A: To simplify an exponential expression with a fraction, you need to apply the rules for exponents. For example, if you have $\frac{a^m}{b^n}$, you can rewrite it as $\frac{a^m}{b^n} = \frac{a^m}{b^n}$.

Conclusion

In conclusion, we have discussed the rules for simplifying exponential expressions, including the power rule, the rule for negative exponents, and the rule for multiplying and dividing exponential expressions. We have also provided examples of how to apply these rules to simplify expressions with variable exponents and fractions.

Final Answer

The final answer is that the rules for simplifying exponential expressions are:

• Power rule: $(a^m)^n = a^{m \cdot n}$
• Rule for negative exponents: $a^{-n} = \frac{1}{a^n}$
• Rule for multiplying exponential expressions: $a^m \cdot a^n = a^{m+n}$
• Rule for dividing exponential expressions: $\frac{a^m}{a^n} = a^{m-n}$

By applying these rules, you can simplify exponential expressions and solve problems involving exponents.