True Or False: $\left(9^9\right) \cdot \left(9^{-20}\right) = 9^{-29}$A. True B. False

by ADMIN 88 views

Introduction

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. In this article, we will delve into the world of exponents and explore the concept of negative exponents. We will examine the statement (99)⋅(9−20)=9−29\left(9^9\right) \cdot \left(9^{-20}\right) = 9^{-29} and determine whether it is true or false.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication of a number. For example, 939^3 can be written as 9⋅9⋅99 \cdot 9 \cdot 9. Exponents can be positive, negative, or zero. A positive exponent indicates that the number is being multiplied by itself a certain number of times. A negative exponent indicates that the reciprocal of the number is being multiplied by itself a certain number of times.

The Rule of Negative Exponents

The rule of negative exponents states that a−n=1ana^{-n} = \frac{1}{a^n}. This means that a negative exponent can be rewritten as a fraction, where the numerator is 1 and the denominator is the number raised to the power of the negative exponent.

Applying the Rule of Negative Exponents

Let's apply the rule of negative exponents to the statement (99)⋅(9−20)\left(9^9\right) \cdot \left(9^{-20}\right). Using the rule, we can rewrite the second term as 1920\frac{1}{9^{20}}. The statement now becomes (99)⋅1920\left(9^9\right) \cdot \frac{1}{9^{20}}.

Simplifying the Expression

To simplify the expression, we can use the rule of exponents, which states that am⋅an=am+na^m \cdot a^n = a^{m+n}. In this case, we have 99⋅9−209^9 \cdot 9^{-20}. Using the rule, we can add the exponents, resulting in 99−20=9−119^{9-20} = 9^{-11}.

Conclusion

In conclusion, the statement (99)⋅(9−20)=9−29\left(9^9\right) \cdot \left(9^{-20}\right) = 9^{-29} is false. Using the rule of negative exponents and the rule of exponents, we simplified the expression to 9−119^{-11}, which is not equal to 9−299^{-29}.

Additional Examples

To further illustrate the concept of negative exponents, let's consider a few additional examples.

Example 1

(24)⋅(2−3)=24−3=21=2\left(2^4\right) \cdot \left(2^{-3}\right) = 2^{4-3} = 2^1 = 2

Example 2

(35)⋅(3−2)=35−2=33=27\left(3^5\right) \cdot \left(3^{-2}\right) = 3^{5-2} = 3^3 = 27

Example 3

(42)⋅(4−1)=42−1=41=4\left(4^2\right) \cdot \left(4^{-1}\right) = 4^{2-1} = 4^1 = 4

Conclusion

In conclusion, the concept of negative exponents is a powerful tool in mathematics, allowing us to simplify complex expressions and solve problems more efficiently. By understanding the rule of negative exponents and the rule of exponents, we can confidently determine whether a statement is true or false.

Final Answer

Q: What is the rule of negative exponents?

A: The rule of negative exponents states that a−n=1ana^{-n} = \frac{1}{a^n}. This means that a negative exponent can be rewritten as a fraction, where the numerator is 1 and the denominator is the number raised to the power of the negative exponent.

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

A: To simplify an expression with a negative exponent, you can use the rule of negative exponents to rewrite the term as a fraction. Then, you can use the rule of exponents to combine the terms.

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

A: A positive exponent indicates that the number is being multiplied by itself a certain number of times. A negative exponent indicates that the reciprocal of the number is being multiplied by itself a certain number of times.

Q: Can you give me an example of how to simplify an expression with a negative exponent?

A: Let's consider the expression (23)⋅(2−2)\left(2^3\right) \cdot \left(2^{-2}\right). Using the rule of negative exponents, we can rewrite the second term as 122\frac{1}{2^2}. The expression now becomes (23)⋅122\left(2^3\right) \cdot \frac{1}{2^2}. Using the rule of exponents, we can add the exponents, resulting in 23−2=21=22^{3-2} = 2^1 = 2.

Q: How do I know when to use the rule of negative exponents?

A: You should use the rule of negative exponents whenever you encounter a negative exponent in an expression. This will allow you to rewrite the term as a fraction and simplify the expression.

Q: Can you give me some examples of expressions with negative exponents?

A: Here are a few examples:

  • (34)⋅(3−1)=34−1=33=27\left(3^4\right) \cdot \left(3^{-1}\right) = 3^{4-1} = 3^3 = 27
  • (22)⋅(2−3)=22−3=2−1=12\left(2^2\right) \cdot \left(2^{-3}\right) = 2^{2-3} = 2^{-1} = \frac{1}{2}
  • (53)⋅(5−2)=53−2=51=5\left(5^3\right) \cdot \left(5^{-2}\right) = 5^{3-2} = 5^1 = 5

Q: What are some common mistakes to avoid when working with negative exponents?

A: Some common mistakes to avoid when working with negative exponents include:

  • Forgetting to use the rule of negative exponents when simplifying an expression
  • Adding or subtracting the exponents incorrectly
  • Not rewriting the negative exponent as a fraction

Q: How do I know if an expression with a negative exponent is true or false?

A: To determine whether an expression with a negative exponent is true or false, you can use the rule of negative exponents and the rule of exponents to simplify the expression. If the simplified expression is equal to the original expression, then the original expression is true. If the simplified expression is not equal to the original expression, then the original expression is false.

Conclusion

In conclusion, negative exponents are a powerful tool in mathematics, allowing us to simplify complex expressions and solve problems more efficiently. By understanding the rule of negative exponents and the rule of exponents, we can confidently determine whether a statement is true or false.