Find The Value Of The Missing Exponent:$\[ \frac{9^{12}}{9^?}=9^{12} \\]

by ADMIN 73 views

=====================================================

Introduction

In mathematics, exponents are a fundamental concept used to represent repeated multiplication of a number. When dealing with exponents, it's essential to understand the rules of exponentiation, including the properties of equality and the rules for simplifying expressions. In this article, we will explore the concept of exponents and how to find the value of a missing exponent in a given equation.

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 \times 9 \times 9. The exponent, in this case, is 3, and it represents the number of times the base (9) is multiplied by itself. Exponents can be positive, negative, or zero, and they can be used to represent very large or very small numbers.

The Rule of Exponentiation

When dealing with exponents, there are several rules that must be followed. One of the most important rules is the rule of exponentiation, which states that when multiplying two numbers with the same base, you can add their exponents. For example, 93×92=93+2=959^3 \times 9^2 = 9^{3+2} = 9^5. This rule is essential for simplifying expressions and solving equations involving exponents.

The Equation

The equation we are given is 9129?=912\frac{9^{12}}{9^?}=9^{12}. Our goal is to find the value of the missing exponent, denoted by the question mark. To do this, we need to apply the rules of exponentiation and simplify the expression.

Simplifying the Expression

To simplify the expression, we can start by applying the rule of exponentiation. When dividing two numbers with the same base, we can subtract their exponents. In this case, we have 9129?=912\frac{9^{12}}{9^?}=9^{12}. We can rewrite this expression as 912−?=9129^{12-?}=9^{12}.

Finding the Missing Exponent

Now that we have simplified the expression, we can find the value of the missing exponent. To do this, we need to equate the exponents on both sides of the equation. We have 12−?=1212-?=12. To solve for the missing exponent, we can add 12 to both sides of the equation, resulting in −?=0-?=0.

Solving for the Missing Exponent

Now that we have −?=0-?=0, we can solve for the missing exponent. To do this, we can multiply both sides of the equation by -1, resulting in ?=0?=0. This means that the missing exponent is 0.

Conclusion

In conclusion, we have found the value of the missing exponent in the given equation. By applying the rules of exponentiation and simplifying the expression, we were able to determine that the missing exponent is 0. This is a fundamental concept in mathematics, and it's essential to understand the rules of exponentiation to solve equations involving exponents.

Frequently Asked Questions

Q: What is the rule of exponentiation?

A: The rule of exponentiation states that when multiplying two numbers with the same base, you can add their exponents.

Q: How do you simplify an expression involving exponents?

A: To simplify an expression involving exponents, you can apply the rule of exponentiation and simplify the expression.

Q: How do you find the value of a missing exponent?

A: To find the value of a missing exponent, you need to equate the exponents on both sides of the equation and solve for the missing exponent.

Final Thoughts

In conclusion, finding the value of a missing exponent is a fundamental concept in mathematics. By understanding the rules of exponentiation and simplifying expressions, we can solve equations involving exponents. This is a critical skill that is essential for success in mathematics and other fields.

Additional Resources

For more information on exponents and how to find the value of a missing exponent, check out the following resources:

  • Khan Academy: Exponents and Exponential Functions
  • Mathway: Exponents and Exponential Functions
  • Wolfram Alpha: Exponents and Exponential Functions

References

  • "Algebra and Trigonometry" by Michael Sullivan
  • "Calculus" by Michael Spivak
  • "Mathematics for the Nonmathematician" by Morris Kline

=====================================================

Introduction

Exponents are a fundamental concept in mathematics, and they can be a bit tricky to understand at first. In this article, we will answer some of the most frequently asked questions about exponents, including how to simplify expressions, how to find the value of a missing exponent, and more.

Q: What is an exponent?

A: An exponent is a small number that is written above and to the right of a base number. It represents the number of times the base number is multiplied by itself. For example, 939^3 means 9×9×99 \times 9 \times 9.

Q: What is the rule of exponentiation?

A: The rule of exponentiation states that when multiplying two numbers with the same base, you can add their exponents. For example, 93×92=93+2=959^3 \times 9^2 = 9^{3+2} = 9^5.

Q: How do you simplify an expression involving exponents?

A: To simplify an expression involving exponents, you can apply the rule of exponentiation and simplify the expression. For example, 91293=912−3=99\frac{9^{12}}{9^3} = 9^{12-3} = 9^9.

Q: How do you find the value of a missing exponent?

A: To find the value of a missing exponent, you need to equate the exponents on both sides of the equation and solve for the missing exponent. For example, 9129?=912\frac{9^{12}}{9^?} = 9^{12} means that 12−?=1212-? = 12, so ?=0? = 0.

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

A: A positive exponent represents a number that is being multiplied by itself a certain number of times. A negative exponent represents a number that is being divided by itself a certain number of times. For example, 939^3 means 9×9×99 \times 9 \times 9, while 9−39^{-3} means 19×9×9\frac{1}{9 \times 9 \times 9}.

Q: How do you handle exponents with different bases?

A: When dealing with exponents with different bases, you can use the rule of exponentiation to simplify the expression. For example, 9343=93−343=9043=143\frac{9^3}{4^3} = \frac{9^{3-3}}{4^3} = \frac{9^0}{4^3} = \frac{1}{4^3}.

Q: Can you explain the concept of zero exponent?

A: Yes, a zero exponent means that the base number is being multiplied by itself zero times, which is equal to 1. For example, 90=19^0 = 1.

Q: Can you explain the concept of negative exponent?

A: Yes, a negative exponent means that the base number is being divided by itself a certain number of times. For example, 9−3=19×9×99^{-3} = \frac{1}{9 \times 9 \times 9}.

Q: How do you handle exponents with fractions?

A: When dealing with exponents with fractions, you can use the rule of exponentiation to simplify the expression. For example, (94)3=9343\left(\frac{9}{4}\right)^3 = \frac{9^3}{4^3}.

Q: Can you explain the concept of exponential functions?

A: Yes, an exponential function is a function that has the form f(x)=axf(x) = a^x, where aa is a positive number and xx is the variable. Exponential functions can be used to model real-world phenomena, such as population growth and chemical reactions.

Q: Can you explain the concept of logarithmic functions?

A: Yes, a logarithmic function is a function that has the form f(x)=log⁡a(x)f(x) = \log_a(x), where aa is a positive number and xx is the variable. Logarithmic functions can be used to model real-world phenomena, such as sound levels and pH levels.

Conclusion

In conclusion, exponents are a fundamental concept in mathematics, and they can be a bit tricky to understand at first. However, with practice and patience, you can become proficient in working with exponents and solving equations involving exponents. We hope that this article has been helpful in answering some of the most frequently asked questions about exponents.

Additional Resources

For more information on exponents and how to solve equations involving exponents, check out the following resources:

  • Khan Academy: Exponents and Exponential Functions
  • Mathway: Exponents and Exponential Functions
  • Wolfram Alpha: Exponents and Exponential Functions

References

  • "Algebra and Trigonometry" by Michael Sullivan
  • "Calculus" by Michael Spivak
  • "Mathematics for the Nonmathematician" by Morris Kline