Simplify $9^4 \times 9^6$.Give Your Answer As A Power Of 9.

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Understanding Exponents and Multiplication

When dealing with exponents, it's essential to understand the rules of multiplication. In this case, we're given the expression $9^4 \times 9^6$. To simplify this expression, we need to apply the rule of multiplication for exponents, which states that when multiplying two numbers with the same base, we add their exponents.

The Rule of Multiplication for Exponents

The rule of multiplication for exponents is as follows:

$a^m \times a^n = a^{m+n}$

where $a$ is the base and $m$ and $n$ are the exponents.

Applying the Rule to the Given Expression

Now, let's apply this rule to the given expression $9^4 \times 9^6$. Since both terms have the same base, 9, we can add their exponents.

$9^4 \times 9^6 = 9^{4+6}$

Simplifying the Expression

Now, let's simplify the expression by evaluating the exponent.

$9^{4+6} = 9^{10}$

Conclusion

Therefore, the simplified expression is $9^{10}$. This is the result of applying the rule of multiplication for exponents to the given expression.

Understanding the Concept of Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $9^4$ can be written as $9 \times 9 \times 9 \times 9$. Exponents make it easier to write and evaluate expressions involving repeated multiplication.

The Importance of Exponents in Mathematics

Exponents play a crucial role in mathematics, particularly in algebra and calculus. They are used to represent complex expressions and to simplify calculations.

Real-World Applications of Exponents

Exponents have numerous real-world applications, including finance, science, and engineering. For example, compound interest is calculated using exponents, and exponential growth is used to model population growth and other phenomena.

Simplifying Expressions with Exponents

Simplifying expressions with exponents involves applying the rules of exponentiation. These rules include the rule of multiplication, the rule of division, and the rule of power.

The Rule of Multiplication

The rule of multiplication states that when multiplying two numbers with the same base, we add their exponents.

$a^m \times a^n = a^{m+n}$

The Rule of Division

The rule of division states that when dividing two numbers with the same base, we subtract their exponents.

$\frac{a^m}{a^n} = a^{m-n}$

The Rule of Power

The rule of power states that when raising a power to a power, we multiply the exponents.

$(a^m)^n = a^{m \times n}$

Applying the Rules to Simplify Expressions

Now, let's apply these rules to simplify some expressions.

Example 1

Simplify the expression $2^3 \times 2^4$.

Using the rule of multiplication, we add the exponents.

$2^3 \times 2^4 = 2^{3+4} = 2^7$

Example 2

Simplify the expression $\frac{2^5}{2^3}$.

Using the rule of division, we subtract the exponents.

$\frac{2^5}{2^3} = 2^{5-3} = 2^2$

Example 3

Simplify the expression $(2^3)^4$.

Using the rule of power, we multiply the exponents.

$(2^3)^4 = 2^{3 \times 4} = 2^{12}$

Conclusion

In conclusion, simplifying expressions with exponents involves applying the rules of exponentiation. These rules include the rule of multiplication, the rule of division, and the rule of power. By applying these rules, we can simplify complex expressions and make calculations easier.

Final Thoughts

In this article, we've discussed the concept of exponents and how to simplify expressions with exponents. We've applied the rules of exponentiation to simplify expressions and have seen the importance of exponents in mathematics. Exponents are a powerful tool that can be used to represent complex expressions and to simplify calculations. By understanding and applying the rules of exponentiation, we can make calculations easier and more efficient.

Key Takeaways

• Exponents are a shorthand way of representing repeated multiplication.
• The rule of multiplication states that when multiplying two numbers with the same base, we add their exponents.
• The rule of division states that when dividing two numbers with the same base, we subtract their exponents.
• The rule of power states that when raising a power to a power, we multiply the exponents.
• Simplifying expressions with exponents involves applying the rules of exponentiation.

Further Reading

For further reading on exponents and simplifying expressions, we recommend the following resources:

• Khan Academy: Exponents and Exponential Functions
• Math Is Fun: Exponents
• Wolfram MathWorld: Exponents

Conclusion

In conclusion, simplifying expressions with exponents is an essential skill in mathematics. By understanding and applying the rules of exponentiation, we can simplify complex expressions and make calculations easier. We hope this article has provided a comprehensive overview of exponents and how to simplify expressions with exponents.

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Frequently Asked Questions

Q: What is the rule of multiplication for exponents?

A: The rule of multiplication for exponents states that when multiplying two numbers with the same base, we add their exponents. This can be represented as:

$a^m \times a^n = a^{m+n}$

Q: How do I apply the rule of multiplication for exponents?

A: To apply the rule of multiplication for exponents, simply add the exponents of the two numbers. For example:

$9^4 \times 9^6 = 9^{4+6} = 9^{10}$

Q: What is the rule of division for exponents?

A: The rule of division for exponents states that when dividing two numbers with the same base, we subtract their exponents. This can be represented as:

$\frac{a^m}{a^n} = a^{m-n}$

Q: How do I apply the rule of division for exponents?

A: To apply the rule of division for exponents, simply subtract the exponents of the two numbers. For example:

$\frac{9^5}{9^3} = 9^{5-3} = 9^2$

Q: What is the rule of power for exponents?

A: The rule of power for exponents states that when raising a power to a power, we multiply the exponents. This can be represented as:

$(a^m)^n = a^{m \times n}$

Q: How do I apply the rule of power for exponents?

A: To apply the rule of power for exponents, simply multiply the exponents of the two numbers. For example:

$(9^3)^4 = 9^{3 \times 4} = 9^{12}$

Q: Can I simplify expressions with exponents that have different bases?

A: No, the rules of exponentiation only apply to expressions with the same base. If the bases are different, you cannot simplify the expression using the rules of exponentiation.

Q: Can I simplify expressions with exponents that have negative exponents?

A: Yes, you can simplify expressions with exponents that have negative exponents. To do this, you can use the rule of negative exponents, which states that:

$a^{-n} = \frac{1}{a^n}$

Q: How do I simplify expressions with exponents that have negative exponents?

A: To simplify expressions with exponents that have negative exponents, you can use the rule of negative exponents. For example:

$9^{-3} = \frac{1}{9^3}$

Q: Can I simplify expressions with exponents that have fractional exponents?

A: Yes, you can simplify expressions with exponents that have fractional exponents. To do this, you can use the rule of fractional exponents, which states that:

$a^{\frac{m}{n}} = \sqrt[n]{a^m}$

Q: How do I simplify expressions with exponents that have fractional exponents?

A: To simplify expressions with exponents that have fractional exponents, you can use the rule of fractional exponents. For example:

$9^{\frac{1}{2}} = \sqrt{9}$

Q: Can I simplify expressions with exponents that have complex exponents?

A: Yes, you can simplify expressions with exponents that have complex exponents. To do this, you can use the rule of complex exponents, which states that:

$a^{m+ni} = a^m(\cos(n\ln(a)) + i\sin(n\ln(a)))$

Q: How do I simplify expressions with exponents that have complex exponents?

A: To simplify expressions with exponents that have complex exponents, you can use the rule of complex exponents. For example:

$9^{3+4i} = 9^3(\cos(4\ln(9)) + i\sin(4\ln(9)))$

Conclusion

In conclusion, simplifying expressions with exponents is an essential skill in mathematics. By understanding and applying the rules of exponentiation, you can simplify complex expressions and make calculations easier. We hope this Q&A article has provided a comprehensive overview of exponents and how to simplify expressions with exponents.

Key Takeaways

• The rule of multiplication for exponents states that when multiplying two numbers with the same base, we add their exponents.
• The rule of division for exponents states that when dividing two numbers with the same base, we subtract their exponents.
• The rule of power for exponents states that when raising a power to a power, we multiply the exponents.
• Simplifying expressions with exponents involves applying the rules of exponentiation.
• Exponents can be simplified using the rules of exponentiation, including the rule of multiplication, the rule of division, and the rule of power.

Further Reading

For further reading on exponents and simplifying expressions, we recommend the following resources:

• Khan Academy: Exponents and Exponential Functions
• Math Is Fun: Exponents
• Wolfram MathWorld: Exponents

Conclusion

In conclusion, simplifying expressions with exponents is an essential skill in mathematics. By understanding and applying the rules of exponentiation, you can simplify complex expressions and make calculations easier. We hope this Q&A article has provided a comprehensive overview of exponents and how to simplify expressions with exponents.