Find The Solution To The Equation Using What You Know About Exponent Rules.$9^? \cdot 9^7 = 9^7$

Introduction

Exponential equations can be challenging to solve, but with a solid understanding of exponent rules, you can find the solution with ease. In this article, we will explore how to solve the equation $9^? \cdot 9^7 = 9^7$ using exponent rules.

Understanding Exponent Rules

Before we dive into solving the equation, let's review some essential exponent rules:

• Product of Powers Rule: When multiplying two powers with the same base, add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power Rule: When raising a power to another power, multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.
• Quotient of Powers Rule: When dividing two powers with the same base, subtract the exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Solving the Equation

Now that we have reviewed the exponent rules, let's apply them to solve the equation $9^? \cdot 9^7 = 9^7$.

Step 1: Apply the Product of Powers Rule

Using the product of powers rule, we can rewrite the equation as:

$$9^{?+7} = 9^7$$

Step 2: Equate the Exponents

Since the bases are the same, we can equate the exponents:

$$?+7 = 7$$

Step 3: Solve for the Unknown Exponent

Now, let's solve for the unknown exponent:

$$? = 7 - 7$$

$$? = 0$$

Step 4: Verify the Solution

To verify the solution, let's substitute the value of the unknown exponent back into the original equation:

$$9^0 \cdot 9^7 = 9^7$$

Using the product of powers rule, we can rewrite the equation as:

$$9^{0+7} = 9^7$$

$$9^7 = 9^7$$

The equation holds true, so our solution is correct.

Conclusion

Solving exponential equations requires a solid understanding of exponent rules. By applying the product of powers rule, equating the exponents, and solving for the unknown exponent, we can find the solution to the equation $9^? \cdot 9^7 = 9^7$. Remember to verify your solution by substituting the value back into the original equation.

Common Mistakes to Avoid

When solving exponential equations, it's essential to avoid common mistakes:

• Not applying the product of powers rule: Failing to apply the product of powers rule can lead to incorrect solutions.
• Not equating the exponents: Failing to equate the exponents can lead to incorrect solutions.
• Not verifying the solution: Failing to verify the solution can lead to incorrect solutions.

Tips and Tricks

Here are some tips and tricks to help you solve exponential equations:

• Use the product of powers rule: The product of powers rule is a powerful tool for solving exponential equations.
• Equate the exponents: Equating the exponents is a crucial step in solving exponential equations.
• Verify the solution: Verifying the solution is essential to ensure that your answer is correct.

Real-World Applications

Exponential equations have numerous real-world applications:

• Finance: Exponential equations are used to model population growth, compound interest, and inflation.
• Science: Exponential equations are used to model chemical reactions, population growth, and radioactive decay.
• Engineering: Exponential equations are used to model electrical circuits, mechanical systems, and thermal systems.

Practice Problems

Here are some practice problems to help you reinforce your understanding of exponential equations:

• Problem 1: Solve the equation $2^? \cdot 2^5 = 2^5$
• Problem 2: Solve the equation $3^? \cdot 3^3 = 3^3$
• Problem 3: Solve the equation $4^? \cdot 4^2 = 4^2$

Conclusion

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Frequently Asked Questions

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential expression, which is a number raised to a power. For example, $2^3$ is an exponential expression.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to apply the exponent rules, such as the product of powers rule, the power of a power rule, and the quotient of powers rule. You also need to equate the exponents and solve for the unknown exponent.

Q: What is the product of powers rule?

A: The product of powers rule states that when multiplying two powers with the same base, you add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.

Q: What is the power of a power rule?

A: The power of a power rule states that when raising a power to another power, you multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.

Q: What is the quotient of powers rule?

A: The quotient of powers rule states that when dividing two powers with the same base, you subtract the exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

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

A: To apply the product of powers rule, you need to multiply the two powers with the same base and add the exponents. For example, $2^3 \cdot 2^4 = 2^{3+4} = 2^7$.

Q: How do I apply the power of a power rule?

A: To apply the power of a power rule, you need to raise the power to another power and multiply the exponents. For example, $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$.

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

A: To apply the quotient of powers rule, you need to divide the two powers with the same base and subtract the exponents. For example, $\frac{2^5}{2^3} = 2^{5-3} = 2^2$.

Q: What are some common mistakes to avoid when solving exponential equations?

A: Some common mistakes to avoid when solving exponential equations include:

• Not applying the product of powers rule
• Not equating the exponents
• Not verifying the solution

Q: How do I verify the solution to an exponential equation?

A: To verify the solution to an exponential equation, you need to substitute the value back into the original equation and check if it holds true.

Q: What are some real-world applications of exponential equations?

A: Exponential equations have numerous real-world applications, including:

• Finance: Exponential equations are used to model population growth, compound interest, and inflation.
• Science: Exponential equations are used to model chemical reactions, population growth, and radioactive decay.
• Engineering: Exponential equations are used to model electrical circuits, mechanical systems, and thermal systems.

Q: How can I practice solving exponential equations?

A: You can practice solving exponential equations by working on practice problems, such as:

• Solving equations with the same base
• Solving equations with different bases
• Solving equations with negative exponents

Conclusion

Exponential equations can be challenging to solve, but with practice and patience, you can become proficient in solving them. Remember to apply the exponent rules, equate the exponents, and verify the solution to ensure that your answer is correct. With this Q&A guide, you can better understand exponential equations and how to solve them.