Simplify $\frac{3^5 \times 3^9}{3^3 \times 3^2}$ Leaving Your Answer In Index Form.

$$

Introduction

In mathematics, simplifying expressions with exponents is a crucial skill that helps us solve complex problems efficiently. One of the most common techniques used to simplify expressions with exponents is the rule of multiplication and division of exponents. In this article, we will focus on simplifying the expression $\frac{3^5 \times 3^9}{3^3 \times 3^2}$ using the rule of multiplication and division of exponents.

Understanding Exponents

Before we dive into simplifying the expression, let's quickly review what exponents are. An exponent is a small number that is written above and to the right of a number or a variable. It represents the power or the index to which the number or variable is raised. For example, in the expression $3^5$, the exponent 5 represents the power to which the number 3 is raised. In other words, $3^5$ is equal to $3 \times 3 \times 3 \times 3 \times 3$.

The Rule of Multiplication and Division of Exponents

The rule of multiplication and division of exponents states that when we multiply two numbers or variables with the same base, we add their exponents. On the other hand, when we divide two numbers or variables with the same base, we subtract their exponents. Mathematically, this can be represented as:

• $a^m \times a^n = a^{m+n}$
• $\frac{a^m}{a^n} = a^{m-n}$

Simplifying the Expression

Now that we have a good understanding of exponents and the rule of multiplication and division of exponents, let's simplify the expression $\frac{3^5 \times 3^9}{3^3 \times 3^2}$.

Using the rule of multiplication and division of exponents, we can simplify the numerator and denominator separately.

• Numerator: $3^5 \times 3^9 = 3^{5+9} = 3^{14}$
• Denominator: $3^3 \times 3^2 = 3^{3+2} = 3^5$

Now that we have simplified the numerator and denominator, we can rewrite the expression as:

$\frac{3^{14}}{3^5}$

Using the rule of division of exponents, we can simplify the expression further by subtracting the exponents:

$\frac{3^{14}}{3^5} = 3^{14-5} = 3^9$

Therefore, the simplified form of the expression $\frac{3^5 \times 3^9}{3^3 \times 3^2}$ is $3^9$.

Conclusion

In this article, we simplified the expression $\frac{3^5 \times 3^9}{3^3 \times 3^2}$ using the rule of multiplication and division of exponents. We reviewed the basics of exponents and the rule of multiplication and division of exponents, and then applied it to simplify the given expression. The simplified form of the expression is $3^9$. This article demonstrates the importance of understanding exponents and the rule of multiplication and division of exponents in simplifying complex expressions.

Example Problems

Here are a few example problems that demonstrate the application of the rule of multiplication and division of exponents:

• $\frac{2^7 \times 2^3}{2^4 \times 2^2} = 2^{7+3-4-2} = 2^4$
• $\frac{5^8 \times 5^2}{5^3 \times 5^5} = 5^{8+2-3-5} = 5^2$
• $\frac{7^9 \times 7^4}{7^2 \times 7^6} = 7^{9+4-2-6} = 7^5$

These example problems demonstrate the application of the rule of multiplication and division of exponents in simplifying complex expressions.

Practice Problems

Here are a few practice problems that you can try to apply the rule of multiplication and division of exponents:

• $\frac{3^6 \times 3^8}{3^4 \times 3^3} = ?$
• $\frac{4^9 \times 4^2}{4^5 \times 4^4} = ?$
• $\frac{6^7 \times 6^3}{6^2 \times 6^5} = ?$

Try to simplify these expressions using the rule of multiplication and division of exponents, and then check your answers with the solutions provided below.

Solutions

Here are the solutions to the practice problems:

• $\frac{3^6 \times 3^8}{3^4 \times 3^3} = 3^{6+8-4-3} = 3^7$
• $\frac{4^9 \times 4^2}{4^5 \times 4^4} = 4^{9+2-5-4} = 4^2$
• $\frac{6^7 \times 6^3}{6^2 \times 6^5} = 6^{7+3-2-5} = 6^3$

Q: What is the rule of multiplication and division of exponents?

A: The rule of multiplication and division of exponents states that when we multiply two numbers or variables with the same base, we add their exponents. On the other hand, when we divide two numbers or variables with the same base, we subtract their exponents. Mathematically, this can be represented as:

• $a^m \times a^n = a^{m+n}$
• $\frac{a^m}{a^n} = a^{m-n}$

Q: How do I simplify an expression with exponents using the rule of multiplication and division of exponents?

A: To simplify an expression with exponents using the rule of multiplication and division of exponents, follow these steps:

1. Identify the base and the exponents in the expression.
2. If the expression involves multiplication, add the exponents.
3. If the expression involves division, subtract the exponents.
4. Simplify the resulting expression.

Q: What is the difference between the rule of multiplication and division of exponents and the rule of addition and subtraction of exponents?

A: The rule of addition and subtraction of exponents states that when we add or subtract exponents with the same base, we add or subtract the exponents directly. However, when we multiply or divide exponents with the same base, we add or subtract the exponents using the rule of multiplication and division of exponents.

Q: Can I simplify an expression with exponents that has a negative exponent?

A: Yes, you can simplify an expression with exponents that has a negative exponent. When simplifying an expression with a negative exponent, remember that a negative exponent represents the reciprocal of the base raised to the positive exponent. For example, $a^{-m} = \frac{1}{a^m}$.

Q: How do I simplify an expression with exponents that has a zero exponent?

A: When simplifying an expression with exponents that has a zero exponent, remember that any number or variable raised to the power of zero is equal to 1. For example, $a^0 = 1$.

Q: Can I simplify an expression with exponents that has a fractional exponent?

A: Yes, you can simplify an expression with exponents that has a fractional exponent. When simplifying an expression with a fractional exponent, remember that the numerator represents the power to which the base is raised, and the denominator represents the root of the base. For example, $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.

Q: How do I simplify an expression with exponents that has a negative fractional exponent?

A: When simplifying an expression with exponents that has a negative fractional exponent, remember that the negative sign in the exponent represents the reciprocal of the base raised to the positive exponent. For example, $a^{-\frac{m}{n}} = \frac{1}{\sqrt[n]{a^m}}$.

Q: Can I simplify an expression with exponents that has a variable in the exponent?

A: Yes, you can simplify an expression with exponents that has a variable in the exponent. When simplifying an expression with a variable in the exponent, remember that the variable represents the power to which the base is raised. For example, $a^{2x} = (a^2)^x$.

Q: How do I simplify an expression with exponents that has a coefficient in the exponent?

A: When simplifying an expression with exponents that has a coefficient in the exponent, remember that the coefficient represents the power to which the base is raised. For example, $2a^3 = (2a)^3$.

Q: Can I simplify an expression with exponents that has a complex number in the exponent?

A: Yes, you can simplify an expression with exponents that has a complex number in the exponent. When simplifying an expression with a complex number in the exponent, remember that the complex number represents the power to which the base is raised. For example, $a^{3+4i} = a^3 \cdot a^{4i}$.

Conclusion

In this article, we have answered some of the most frequently asked questions on simplifying expressions with exponents. We have covered topics such as the rule of multiplication and division of exponents, simplifying expressions with negative exponents, fractional exponents, and variables in the exponent. We hope that this article has helped you understand the basics of simplifying expressions with exponents and has provided you with the confidence to tackle more complex problems.