Simplify The Expression. Write Your Answer Without Using Exponents.$\[ 8^{-1} \cdot 8^{-5} \cdot 8^3 \\]Enter The Correct Answer In The Box.

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Understanding Exponents and Negative Numbers

When dealing with exponents, it's essential to understand the rules and properties that govern them. In this article, we'll focus on simplifying an expression involving negative exponents and explore the concept of reciprocals.

What are Negative Exponents?

A negative exponent is a shorthand way of expressing a fraction. For example, 8βˆ’18^{-1} can be written as 18\frac{1}{8}. This is because 8βˆ’18^{-1} means "8 to the power of -1," which is equivalent to the reciprocal of 8.

Reciprocals and Negative Exponents

Reciprocals are numbers that are equal to 1 divided by the original number. For example, the reciprocal of 8 is 18\frac{1}{8}. When we see a negative exponent, we can rewrite it as a fraction by taking the reciprocal of the base number.

Simplifying the Expression

Now that we've covered the basics of negative exponents and reciprocals, let's simplify the given expression:

8βˆ’1β‹…8βˆ’5β‹…83{ 8^{-1} \cdot 8^{-5} \cdot 8^3 }

To simplify this expression, we can start by rewriting the negative exponents as fractions:

18β‹…185β‹…83{ \frac{1}{8} \cdot \frac{1}{8^5} \cdot 8^3 }

Next, we can combine the fractions by multiplying the numerators and denominators:

1β‹…18β‹…85β‹…83{ \frac{1 \cdot 1}{8 \cdot 8^5} \cdot 8^3 }

Now, we can simplify the denominator by using the rule that amβ‹…an=am+na^m \cdot a^n = a^{m+n}:

181+5β‹…83{ \frac{1}{8^{1+5}} \cdot 8^3 }

This simplifies to:

186β‹…83{ \frac{1}{8^6} \cdot 8^3 }

Using the Rule of Exponents

Now that we have the expression in a simplified form, we can use the rule of exponents to combine the terms. The rule states that when we multiply two numbers with the same base, we can add their exponents:

186β‹…83=83βˆ’6{ \frac{1}{8^6} \cdot 8^3 = 8^{3-6} }

This simplifies to:

8βˆ’3{ 8^{-3} }

Rewriting the Negative Exponent

As we discussed earlier, a negative exponent can be rewritten as a fraction by taking the reciprocal of the base number. In this case, we can rewrite 8βˆ’38^{-3} as:

183{ \frac{1}{8^3} }

Simplifying the Fraction

To simplify the fraction, we can calculate the value of 838^3:

83=8β‹…8β‹…8=512{ 8^3 = 8 \cdot 8 \cdot 8 = 512 }

Now, we can rewrite the fraction as:

1512{ \frac{1}{512} }

Conclusion

In this article, we simplified an expression involving negative exponents and explored the concept of reciprocals. We learned how to rewrite negative exponents as fractions and used the rule of exponents to combine the terms. By following these steps, we arrived at the final answer:

1512{ \frac{1}{512} }

Key Takeaways

  • Negative exponents can be rewritten as fractions by taking the reciprocal of the base number.
  • The rule of exponents states that when we multiply two numbers with the same base, we can add their exponents.
  • To simplify an expression involving negative exponents, we can use the rule of exponents and rewrite the negative exponent as a fraction.

Practice Problems

  1. Simplify the expression: 2βˆ’2β‹…2βˆ’3β‹…242^{-2} \cdot 2^{-3} \cdot 2^4
  2. Rewrite the negative exponent 3βˆ’23^{-2} as a fraction.
  3. Simplify the expression: 5βˆ’1β‹…5βˆ’2β‹…535^{-1} \cdot 5^{-2} \cdot 5^3

Answer Key

  1. 14\frac{1}{4}
  2. 19\frac{1}{9}
  3. 120\frac{1}{20}
    Simplify the Expression: Q&A =============================

Frequently Asked Questions

In this article, we'll answer some of the most common questions related to simplifying expressions involving negative exponents.

Q: What is a negative exponent?

A: A negative exponent is a shorthand way of expressing a fraction. For example, 8βˆ’18^{-1} can be written as 18\frac{1}{8}. This is because 8βˆ’18^{-1} means "8 to the power of -1," which is equivalent to the reciprocal of 8.

Q: How do I rewrite a negative exponent as a fraction?

A: To rewrite a negative exponent as a fraction, take the reciprocal of the base number. For example, 8βˆ’18^{-1} can be rewritten as 18\frac{1}{8}.

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. For example, amβ‹…an=am+na^m \cdot a^n = a^{m+n}.

Q: How do I simplify an expression involving negative exponents?

A: To simplify an expression involving negative exponents, follow these steps:

  1. Rewrite the negative exponents as fractions by taking the reciprocal of the base number.
  2. Use the rule of exponents to combine the terms.
  3. Simplify the resulting expression.

Q: What is the difference between a negative exponent and a positive exponent?

A: A negative exponent is equivalent to the reciprocal of the base number, while a positive exponent is the base number raised to a power. For example, 8βˆ’18^{-1} is equivalent to 18\frac{1}{8}, while 818^1 is equivalent to 8.

Q: Can I simplify an expression involving negative exponents using a calculator?

A: Yes, you can simplify an expression involving negative exponents using a calculator. However, it's essential to understand the underlying math concepts to ensure accurate results.

Q: How do I handle expressions with multiple negative exponents?

A: To handle expressions with multiple negative exponents, follow these steps:

  1. Rewrite each negative exponent as a fraction by taking the reciprocal of the base number.
  2. Use the rule of exponents to combine the terms.
  3. Simplify the resulting expression.

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

A: No, the rule of exponents only applies to expressions with the same base. If you have an expression with different bases, you'll need to use other math concepts to simplify it.

Q: What are some common mistakes to avoid when simplifying expressions involving negative exponents?

A: Some common mistakes to avoid when simplifying expressions involving negative exponents include:

  • Forgetting to rewrite negative exponents as fractions
  • Misapplying the rule of exponents
  • Failing to simplify the resulting expression

Conclusion

In this article, we answered some of the most common questions related to simplifying expressions involving negative exponents. By understanding the underlying math concepts and following the steps outlined above, you'll be able to simplify expressions involving negative exponents with confidence.

Practice Problems

  1. Simplify the expression: 2βˆ’2β‹…2βˆ’3β‹…242^{-2} \cdot 2^{-3} \cdot 2^4
  2. Rewrite the negative exponent 3βˆ’23^{-2} as a fraction.
  3. Simplify the expression: 5βˆ’1β‹…5βˆ’2β‹…535^{-1} \cdot 5^{-2} \cdot 5^3

Answer Key

  1. 14\frac{1}{4}
  2. 19\frac{1}{9}
  3. 120\frac{1}{20}