Simplify The Expression: ${\frac{8 7}{8 {-3}}=}$

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Introduction

Understanding Exponents and Negative Exponents In mathematics, exponents are a shorthand way of representing repeated multiplication. For example, $8^3$ means $8$ multiplied by itself $3$ times, or $8 \times 8 \times 8$. When we have a negative exponent, it means we are dealing with the reciprocal of the base raised to the positive exponent. In this case, $8^{-3}$ means $\frac{1}{8^3}$.

Simplifying the Expression

To simplify the expression $\frac{8^7}{8^{-3}}$, we can use the rule for dividing exponents with the same base. This rule states that when we divide two powers with the same base, we subtract the exponents. In this case, we have:

$$\frac{8^7}{8^{-3}} = 8^{7-(-3)}$$

Applying the Rule for Negative Exponents

When we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. In this case, we can rewrite $8^{-3}$ as $\frac{1}{8^3}$. However, we can also apply the rule for negative exponents, which states that $a^{-n} = \frac{1}{a^n}$. Therefore, we can rewrite the expression as:

$$\frac{8^7}{8^{-3}} = 8^{7-(-3)} = 8^{7+3}$$

Simplifying the Exponent

Now that we have $8^{7+3}$, we can simplify the exponent by adding the numbers. This gives us:

$$8^{7+3} = 8^10$$

Conclusion

In conclusion, we have simplified the expression $\frac{8^7}{8^{-3}}$ to $8^{10}$. This is the final answer.

Additional Examples

• Simplifying Exponents with the Same Base: $\frac{2^5}{2^3} = 2^{5-3} = 2^2$
• Simplifying Exponents with Different Bases: $\frac{3^4}{2^2} = \frac{3^4}{(2^2)^2} = \frac{3^4}{2^4} = (3^4)^{-1} \times (2^4)^1 = (3^{-4}) \times (2^4)$

Tips and Tricks

• Understanding Exponents: Exponents are a shorthand way of representing repeated multiplication. For example, $8^3$ means $8$ multiplied by itself $3$ times, or $8 \times 8 \times 8$.
• Negative Exponents: When we have a negative exponent, it means we are dealing with the reciprocal of the base raised to the positive exponent. In this case, $8^{-3}$ means $\frac{1}{8^3}$.
• Dividing Exponents: When we divide two powers with the same base, we subtract the exponents. In this case, we have $8^{7-(-3)}$.

Frequently Asked Questions

• What is the rule for dividing exponents with the same base?
  • The rule for dividing exponents with the same base states that when we divide two powers with the same base, we subtract the exponents.
• How do we simplify exponents with the same base?
  • We can simplify exponents with the same base by subtracting the exponents.
• How do we simplify exponents with different bases?
  • We can simplify exponents with different bases by taking the reciprocal of the base and simplifying the exponent.

Conclusion

In conclusion, we have simplified the expression $\frac{8^7}{8^{-3}}$ to $8^{10}$. This is the final answer. We have also discussed the rule for dividing exponents with the same base, and how to simplify exponents with the same base and different bases.

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Q&A: Simplifying Exponents and Negative Exponents

Q: What is the rule for dividing exponents with the same base?

A: The rule for dividing exponents with the same base states that when we divide two powers with the same base, we subtract the exponents. In this case, we have $8^{7-(-3)}$.

Q: How do we simplify exponents with the same base?

A: We can simplify exponents with the same base by subtracting the exponents. For example, $\frac{2^5}{2^3} = 2^{5-3} = 2^2$.

Q: How do we simplify exponents with different bases?

A: We can simplify exponents with different bases by taking the reciprocal of the base and simplifying the exponent. For example, $\frac{3^4}{2^2} = \frac{3^4}{(2^2)^2} = \frac{3^4}{2^4} = (3^4)^{-1} \times (2^4)^1 = (3^{-4}) \times (2^4)$.

Q: What is the rule for negative exponents?

A: When we have a negative exponent, it means we are dealing with the reciprocal of the base raised to the positive exponent. In this case, $8^{-3}$ means $\frac{1}{8^3}$.

Q: How do we simplify negative exponents?

A: We can simplify negative exponents by taking the reciprocal of the base and simplifying the exponent. For example, $8^{-3} = \frac{1}{8^3}$.

Q: Can we simplify exponents with the same base and different bases?

A: Yes, we can simplify exponents with the same base and different bases by using the rules for dividing exponents and negative exponents.

Q: What is the final answer to the expression $\frac{8^7}{8^{-3}}$?

A: The final answer to the expression $\frac{8^7}{8^{-3}}$ is $8^{10}$.

Q: Can you provide more examples of simplifying exponents and negative exponents?

A: Here are some additional examples:

• Simplifying Exponents with the Same Base: $\frac{2^5}{2^3} = 2^{5-3} = 2^2$
• Simplifying Exponents with Different Bases: $\frac{3^4}{2^2} = \frac{3^4}{(2^2)^2} = \frac{3^4}{2^4} = (3^4)^{-1} \times (2^4)^1 = (3^{-4}) \times (2^4)$
• Simplifying Negative Exponents: $8^{-3} = \frac{1}{8^3}$

Q: What are some tips and tricks for simplifying exponents and negative exponents?

A: Here are some tips and tricks for simplifying exponents and negative exponents:

• Understanding Exponents: Exponents are a shorthand way of representing repeated multiplication. For example, $8^3$ means $8$ multiplied by itself $3$ times, or $8 \times 8 \times 8$.
• Negative Exponents: When we have a negative exponent, it means we are dealing with the reciprocal of the base raised to the positive exponent. In this case, $8^{-3}$ means $\frac{1}{8^3}$.
• Dividing Exponents: When we divide two powers with the same base, we subtract the exponents. In this case, we have $8^{7-(-3)}$.

Q: Can you provide more information on the rule for dividing exponents with the same base?

A: The rule for dividing exponents with the same base states that when we divide two powers with the same base, we subtract the exponents. This rule can be applied to any two powers with the same base, as long as the exponents are the same.

Q: Can you provide more information on the rule for negative exponents?

A: The rule for negative exponents states that when we have a negative exponent, it means we are dealing with the reciprocal of the base raised to the positive exponent. This rule can be applied to any power with a negative exponent.

Q: Can you provide more information on simplifying exponents with the same base and different bases?

A: We can simplify exponents with the same base and different bases by using the rules for dividing exponents and negative exponents. For example, $\frac{3^4}{2^2} = \frac{3^4}{(2^2)^2} = \frac{3^4}{2^4} = (3^4)^{-1} \times (2^4)^1 = (3^{-4}) \times (2^4)$.

Q: Can you provide more information on simplifying negative exponents?

A: We can simplify negative exponents by taking the reciprocal of the base and simplifying the exponent. For example, $8^{-3} = \frac{1}{8^3}$.

Q: Can you provide more information on the final answer to the expression $\frac{8^7}{8^{-3}}$?

A: The final answer to the expression $\frac{8^7}{8^{-3}}$ is $8^{10}$. This is the result of simplifying the expression using the rules for dividing exponents and negative exponents.