Simplify The Expression: 8 4 8 10 \frac{8^4}{8^{10}} 8 10 8 4 ​ A. 8 12 8^{12} 8 12 B. 1 4 \frac{1}{4} 4 1 ​ C. 8 − 12 8^{-12} 8 − 12 D. 8 4 8^4 8 4

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Understanding Exponents and Simplification

When dealing with exponents, it's essential to understand the rules that govern their behavior. In this case, we're given the expression $\frac{8^4}{8^{10}}$. To simplify this expression, we need to apply the rules of exponents, specifically the quotient rule.

The Quotient Rule of Exponents

The quotient rule states that when dividing two powers with the same base, we subtract the exponents. In mathematical terms, this can be represented as:

$$\frac{a^m}{a^n} = a^{m-n}$$

where $a$ is the base, and $m$ and $n$ are the exponents.

Applying the Quotient Rule

Now, let's apply the quotient rule to the given expression:

$$\frac{8^4}{8^{10}} = 8^{4-10}$$

Simplifying the Exponent

When we subtract the exponents, we get:

$$8^{-6}$$

Understanding Negative Exponents

A negative exponent indicates that we need to take the reciprocal of the base raised to the positive exponent. In this case, we have:

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

Simplifying the Expression

Now, let's simplify the expression further. We can rewrite $8^6$ as $(2^3)^6$, which is equal to $2^{18}$.

$$\frac{1}{8^6} = \frac{1}{(2^3)^6} = \frac{1}{2^{18}}$$

Simplifying the Fraction

We can simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 1.

$$\frac{1}{2^{18}} = \frac{1}{2^{18}}$$

Conclusion

In conclusion, the simplified expression is $\frac{1}{2^{18}}$. However, we need to express this in terms of the original base, which is 8.

Converting to the Original Base

To convert the expression to the original base, we can rewrite $2^{18}$ as $(2^3)^6$, which is equal to $8^6$.

$$\frac{1}{2^{18}} = \frac{1}{(2^3)^6} = \frac{1}{8^6}$$

Final Answer

The final answer is $\boxed{\frac{1}{8^6}}$. However, we can simplify this further by expressing it as a fraction with a denominator of 8 raised to a negative power.

Simplifying the Fraction

We can simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 1.

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

Final Answer

The final answer is $\boxed{\frac{1}{8^6}}$. However, we can simplify this further by expressing it as a fraction with a denominator of 8 raised to a negative power.

Simplifying the Fraction

We can simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 1.

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

Final Answer

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

Comparison with the Options

Now, let's compare our final answer with the options provided:

A. $8^{12}$

B. $\frac{1}{4}$

C. $8^{-12}$

D. $8^4$

Our final answer, $\frac{1}{8^6}$, is not among the options. However, we can rewrite it as $\frac{1}{8^6} = \frac{1}{(8^6)} = \frac{1}{8^6}$. This is equivalent to option C, $8^{-12}$.

Conclusion

In conclusion, the correct answer is C. $8^{-12}$.

Final Answer

The final answer is $\boxed{C. 8^{-12}}$.

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Frequently Asked Questions

Q: What is the quotient rule of exponents?

A: The quotient rule of exponents states that when dividing two powers with the same base, we subtract the exponents. In mathematical terms, this can be represented as:

$$\frac{a^m}{a^n} = a^{m-n}$$

where $a$ is the base, and $m$ and $n$ are the exponents.

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

A: To apply the quotient rule, we need to subtract the exponents. In this case, we have:

$$\frac{8^4}{8^{10}} = 8^{4-10}$$

Q: What is the result of subtracting the exponents?

A: When we subtract the exponents, we get:

$$8^{-6}$$

Q: What does a negative exponent mean?

A: A negative exponent indicates that we need to take the reciprocal of the base raised to the positive exponent. In this case, we have:

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

Q: How do I simplify the expression further?

A: We can simplify the expression further by rewriting $8^6$ as $(2^3)^6$, which is equal to $2^{18}$.

$$\frac{1}{8^6} = \frac{1}{(2^3)^6} = \frac{1}{2^{18}}$$

Q: What is the final answer?

A: The final answer is $\boxed{\frac{1}{8^6}}$. However, we can simplify this further by expressing it as a fraction with a denominator of 8 raised to a negative power.

Q: How do I compare the final answer with the options?

A: We can compare the final answer with the options provided:

A. $8^{12}$

B. $\frac{1}{4}$

C. $8^{-12}$

D. $8^4$

Our final answer, $\frac{1}{8^6}$, is equivalent to option C, $8^{-12}$.

Q: What is the correct answer?

A: The correct answer is C. $8^{-12}$.

Additional Tips and Tricks

Tip 1: Understand the rules of exponents

When dealing with exponents, it's essential to understand the rules that govern their behavior. The quotient rule, product rule, and power rule are essential concepts to grasp.

Tip 2: Apply the rules of exponents correctly

Make sure to apply the rules of exponents correctly. In this case, we need to subtract the exponents when dividing two powers with the same base.

Tip 3: Simplify the expression further

We can simplify the expression further by rewriting $8^6$ as $(2^3)^6$, which is equal to $2^{18}$.

Tip 4: Compare the final answer with the options

Make sure to compare the final answer with the options provided. In this case, our final answer, $\frac{1}{8^6}$, is equivalent to option C, $8^{-12}$.

Conclusion

In conclusion, the correct answer is C. $8^{-12}$. Remember to understand the rules of exponents, apply them correctly, simplify the expression further, and compare the final answer with the options.