Evaluate The Expression $9^7$ Using The Quotient Rule.

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Introduction

In mathematics, the quotient rule is a fundamental concept used to simplify complex expressions by breaking them down into smaller, more manageable parts. This rule is particularly useful when dealing with exponents and powers. In this article, we will explore how to evaluate the expression $9^7$ using the quotient rule.

Understanding the Quotient Rule

The quotient rule states that for any two numbers a and b, and any integer n, the following equation holds:

(ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}

This rule allows us to simplify expressions by separating the numerator and denominator and raising each part to the power of n.

Applying the Quotient Rule to $9^7$

To evaluate the expression $9^7$ using the quotient rule, we can rewrite it as:

97=(91)79^7 = \left(9^1\right)^7

Using the quotient rule, we can simplify this expression as:

(91)7=9717\left(9^1\right)^7 = \frac{9^7}{1^7}

Simplifying the Expression

Now that we have rewritten the expression using the quotient rule, we can simplify it further. Since $1^7 = 1$, we can cancel out the 1 in the denominator, leaving us with:

9717=97\frac{9^7}{1^7} = 9^7

Evaluating $9^7$

Now that we have simplified the expression, we can evaluate $9^7$ by raising 9 to the power of 7:

97=9×9×9×9×9×9×99^7 = 9 \times 9 \times 9 \times 9 \times 9 \times 9 \times 9

Calculating the Value

To calculate the value of $9^7$, we can multiply 9 by itself 7 times:

9×9=819 \times 9 = 81

81×9=72981 \times 9 = 729

729×9=6561729 \times 9 = 6561

6561×9=590496561 \times 9 = 59049

59049×9=53144159049 \times 9 = 531441

531441×9=4762969531441 \times 9 = 4762969

Conclusion

In conclusion, we have successfully evaluated the expression $9^7$ using the quotient rule. By rewriting the expression as $\left(91\right)7$ and simplifying it using the quotient rule, we were able to cancel out the 1 in the denominator and evaluate the expression as $9^7$. We then calculated the value of $9^7$ by multiplying 9 by itself 7 times.

Frequently Asked Questions

Q: What is the quotient rule in mathematics?

A: The quotient rule is a fundamental concept in mathematics that allows us to simplify complex expressions by breaking them down into smaller, more manageable parts.

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

A: To apply the quotient rule, you can rewrite the expression as $\left(\frac{a}{b}\right)^n$ and simplify it using the rule $\left(\frac{a}{b}\right)^n = \frac{an}{bn}$

Q: Can I use the quotient rule to evaluate any expression?

A: Yes, the quotient rule can be used to evaluate any expression that can be rewritten in the form $\left(\frac{a}{b}\right)^n$

Q: What is the value of $9^7$?

A: The value of $9^7$ is 4782969.

References

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources on the topic.

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Introduction

In our previous article, we explored how to evaluate the expression $9^7$ using the quotient rule. In this article, we will delve deeper into the quotient rule and provide a comprehensive Q&A guide to help you understand and apply this fundamental concept in mathematics.

Q&A: Evaluating Expressions Using the Quotient Rule

Q: What is the quotient rule in mathematics?

A: The quotient rule is a fundamental concept in mathematics that allows us to simplify complex expressions by breaking them down into smaller, more manageable parts. It states that for any two numbers a and b, and any integer n, the following equation holds:

(ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}

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

A: To apply the quotient rule, you can rewrite the expression as $\left(\frac{a}{b}\right)^n$ and simplify it using the rule $\left(\frac{a}{b}\right)^n = \frac{an}{bn}$

Q: Can I use the quotient rule to evaluate any expression?

A: Yes, the quotient rule can be used to evaluate any expression that can be rewritten in the form $\left(\frac{a}{b}\right)^n$

Q: What are some common applications of the quotient rule?

A: The quotient rule has numerous applications in mathematics, including:

  • Simplifying complex fractions
  • Evaluating expressions with exponents
  • Solving equations with variables in the denominator
  • Calculating probabilities and statistics

Q: How do I simplify a complex fraction using the quotient rule?

A: To simplify a complex fraction using the quotient rule, you can rewrite the fraction as $\left(\frac{a}{b}\right)^n$ and simplify it using the rule $\left(\frac{a}{b}\right)^n = \frac{an}{bn}$

Q: Can I use the quotient rule to evaluate expressions with negative exponents?

A: Yes, the quotient rule can be used to evaluate expressions with negative exponents. For example:

(ab)−n=bnan\left(\frac{a}{b}\right)^{-n} = \frac{b^n}{a^n}

Q: How do I evaluate an expression with a variable in the denominator using the quotient rule?

A: To evaluate an expression with a variable in the denominator using the quotient rule, you can rewrite the expression as $\left(\frac{a}{b}\right)^n$ and simplify it using the rule $\left(\frac{a}{b}\right)^n = \frac{an}{bn}$

Q: Can I use the quotient rule to solve equations with variables in the denominator?

A: Yes, the quotient rule can be used to solve equations with variables in the denominator. For example:

x2x3=1x\frac{x^2}{x^3} = \frac{1}{x}

Q: What are some common mistakes to avoid when using the quotient rule?

A: Some common mistakes to avoid when using the quotient rule include:

  • Not rewriting the expression in the correct form
  • Not simplifying the expression correctly
  • Not considering the sign of the exponent

Conclusion

In conclusion, the quotient rule is a powerful tool in mathematics that allows us to simplify complex expressions by breaking them down into smaller, more manageable parts. By understanding and applying the quotient rule, you can evaluate expressions with ease and solve equations with variables in the denominator.

Frequently Asked Questions

Q: What is the quotient rule in mathematics?

A: The quotient rule is a fundamental concept in mathematics that allows us to simplify complex expressions by breaking them down into smaller, more manageable parts.

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

A: To apply the quotient rule, you can rewrite the expression as $\left(\frac{a}{b}\right)^n$ and simplify it using the rule $\left(\frac{a}{b}\right)^n = \frac{an}{bn}$

Q: Can I use the quotient rule to evaluate any expression?

A: Yes, the quotient rule can be used to evaluate any expression that can be rewritten in the form $\left(\frac{a}{b}\right)^n$

References

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources on the topic.