Simplify The Expression:${ 9^7 \cdot 9^6 \cdot 9^2 = }$

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Introduction

When dealing with exponents, it's essential to understand the rules of exponentiation to simplify complex expressions. In this article, we will explore how to simplify the expression 97⋅96⋅929^7 \cdot 9^6 \cdot 9^2 using the properties of exponents.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, 939^3 means 9⋅9⋅99 \cdot 9 \cdot 9. When we multiply numbers with the same base, we can add their exponents. This is known as the product of powers rule.

Product of Powers Rule

The product of powers rule states that when we multiply numbers with the same base, we can add their exponents. Mathematically, this can be represented as:

am⋅an=am+na^m \cdot a^n = a^{m+n}

where aa is the base and mm and nn are the exponents.

Simplifying the Expression

Using the product of powers rule, we can simplify the expression 97⋅96⋅929^7 \cdot 9^6 \cdot 9^2 as follows:

97⋅96⋅92=97+6+29^7 \cdot 9^6 \cdot 9^2 = 9^{7+6+2}

Applying the Product of Powers Rule

Now, we can apply the product of powers rule to add the exponents:

97+6+2=9159^{7+6+2} = 9^{15}

Conclusion

In conclusion, we have simplified the expression 97⋅96⋅929^7 \cdot 9^6 \cdot 9^2 using the product of powers rule. By adding the exponents, we arrived at the simplified expression 9159^{15}.

Additional Examples

To further illustrate the concept, let's consider a few more examples:

  • 23⋅24=23+4=272^3 \cdot 2^4 = 2^{3+4} = 2^7
  • 32⋅35=32+5=373^2 \cdot 3^5 = 3^{2+5} = 3^7
  • 46⋅43=46+3=494^6 \cdot 4^3 = 4^{6+3} = 4^9

Real-World Applications

Understanding the product of powers rule has numerous real-world applications. For instance, in finance, it can be used to calculate compound interest. In science, it can be used to model population growth and decay.

Tips and Tricks

Here are a few tips and tricks to help you simplify expressions using the product of powers rule:

  • Make sure the bases are the same before applying the rule.
  • Add the exponents carefully to avoid errors.
  • Use the rule to simplify complex expressions step by step.

Common Mistakes

When simplifying expressions using the product of powers rule, it's essential to avoid common mistakes. Here are a few to watch out for:

  • Not adding the exponents correctly.
  • Not checking if the bases are the same before applying the rule.
  • Not simplifying the expression step by step.

Final Thoughts

In conclusion, the product of powers rule is a powerful tool for simplifying complex expressions. By understanding and applying this rule, you can simplify expressions with ease and accuracy. Remember to always check if the bases are the same before applying the rule and to add the exponents carefully to avoid errors.

Frequently Asked Questions

Q: What is the product of powers rule?

A: The product of powers rule states that when we multiply numbers with the same base, we can add their exponents.

Q: How do I apply the product of powers rule?

A: To apply the product of powers rule, make sure the bases are the same, add the exponents, and simplify the expression.

Q: What are some real-world applications of the product of powers rule?

A: The product of powers rule has numerous real-world applications, including calculating compound interest and modeling population growth and decay.

Q: What are some common mistakes to avoid when simplifying expressions using the product of powers rule?

A: Some common mistakes to avoid include not adding the exponents correctly, not checking if the bases are the same before applying the rule, and not simplifying the expression step by step.

References

Related Topics

  • [1] Exponents and Exponential Functions
  • [2] Algebraic Expressions and Equations
  • [3] Mathematical Modeling and Applications

Introduction

In our previous article, we explored how to simplify the expression 97⋅96⋅929^7 \cdot 9^6 \cdot 9^2 using the product of powers rule. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is the product of powers rule?

A: The product of powers rule states that when we multiply numbers with the same base, we can add their exponents.

Q: How do I apply the product of powers rule?

A: To apply the product of powers rule, make sure the bases are the same, add the exponents, and simplify the expression.

Q: What are some examples of the product of powers rule?

A: Here are a few examples:

  • 23⋅24=23+4=272^3 \cdot 2^4 = 2^{3+4} = 2^7
  • 32⋅35=32+5=373^2 \cdot 3^5 = 3^{2+5} = 3^7
  • 46⋅43=46+3=494^6 \cdot 4^3 = 4^{6+3} = 4^9

Q: What are some real-world applications of the product of powers rule?

A: The product of powers rule has numerous real-world applications, including calculating compound interest and modeling population growth and decay.

Q: What are some common mistakes to avoid when simplifying expressions using the product of powers rule?

A: Some common mistakes to avoid include not adding the exponents correctly, not checking if the bases are the same before applying the rule, and not simplifying the expression step by step.

Q: Can I use the product of powers rule with negative exponents?

A: Yes, you can use the product of powers rule with negative exponents. For example:

  • 2−3⋅2−4=2(−3)+(−4)=2−72^{-3} \cdot 2^{-4} = 2^{(-3)+(-4)} = 2^{-7}

Q: Can I use the product of powers rule with fractional exponents?

A: Yes, you can use the product of powers rule with fractional exponents. For example:

  • 21/2⋅21/3=2(1/2)+(1/3)=25/62^{1/2} \cdot 2^{1/3} = 2^{(1/2)+(1/3)} = 2^{5/6}

Q: How do I simplify expressions with multiple bases?

A: To simplify expressions with multiple bases, you can use the product of powers rule separately for each base. For example:

  • 23⋅32⋅46=(23)⋅(32)⋅(46)=23+0+0⋅30+2+0⋅40+0+6=23⋅32⋅462^3 \cdot 3^2 \cdot 4^6 = (2^3) \cdot (3^2) \cdot (4^6) = 2^{3+0+0} \cdot 3^{0+2+0} \cdot 4^{0+0+6} = 2^3 \cdot 3^2 \cdot 4^6

Q: Can I use the product of powers rule with variables as bases?

A: Yes, you can use the product of powers rule with variables as bases. For example:

  • x2⋅x3=x2+3=x5x^2 \cdot x^3 = x^{2+3} = x^5

Q: How do I simplify expressions with exponents and fractions?

A: To simplify expressions with exponents and fractions, you can use the product of powers rule and the rules for fractions. For example:

  • (23)1/2=23⋅(1/2)=23/2(2^3)^{1/2} = 2^{3 \cdot (1/2)} = 2^{3/2}

Q: Can I use the product of powers rule with complex numbers as bases?

A: Yes, you can use the product of powers rule with complex numbers as bases. For example:

  • (2+3i)2⋅(2+3i)3=(2+3i)2+3=(2+3i)5(2+3i)^2 \cdot (2+3i)^3 = (2+3i)^{2+3} = (2+3i)^5

Conclusion

In conclusion, the product of powers rule is a powerful tool for simplifying complex expressions. By understanding and applying this rule, you can simplify expressions with ease and accuracy. Remember to always check if the bases are the same before applying the rule and to add the exponents carefully to avoid errors.

Frequently Asked Questions

Q: What is the product of powers rule?

A: The product of powers rule states that when we multiply numbers with the same base, we can add their exponents.

Q: How do I apply the product of powers rule?

A: To apply the product of powers rule, make sure the bases are the same, add the exponents, and simplify the expression.

Q: What are some real-world applications of the product of powers rule?

A: The product of powers rule has numerous real-world applications, including calculating compound interest and modeling population growth and decay.

Q: What are some common mistakes to avoid when simplifying expressions using the product of powers rule?

A: Some common mistakes to avoid include not adding the exponents correctly, not checking if the bases are the same before applying the rule, and not simplifying the expression step by step.

References

Related Topics

  • [1] Exponents and Exponential Functions
  • [2] Algebraic Expressions and Equations
  • [3] Mathematical Modeling and Applications