Simplify The Expression:${ \frac{a 3}{a {12}} \times \left(a 3\right) 3 }$

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will delve into the world of algebraic manipulation and explore the steps involved in simplifying a complex expression. We will focus on the given expression: $\frac{a^3}{a^{12}} \times \left(a^3\right)^3$. Our goal is to simplify this expression using the rules of exponents and algebraic manipulation.

Understanding the Rules of Exponents

Before we dive into the simplification process, it's essential to understand the rules of exponents. The rules of exponents state that when we multiply two numbers with the same base, we add their exponents. For example, $a^m \times a^n = a^{m+n}$. Similarly, when we divide two numbers with the same base, we subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Simplifying the Expression

Now that we have a solid understanding of the rules of exponents, let's simplify the given expression. The expression is $\frac{a^3}{a^{12}} \times \left(a^3\right)^3$. To simplify this expression, we will follow the order of operations (PEMDAS):

1. Evaluate the exponentiation: The first step is to evaluate the exponentiation. In this case, we have $\left(a^3\right)^3$. Using the rule of exponents, we can rewrite this as $a^{3 \times 3} = a^9$.
2. Simplify the fraction: Now that we have evaluated the exponentiation, let's simplify the fraction. We have $\frac{a^3}{a^{12}}$. Using the rule of exponents, we can rewrite this as $a^{3-12} = a^{-9}$.
3. Multiply the terms: Finally, let's multiply the terms. We have $a^{-9} \times a^9$. Using the rule of exponents, we can rewrite this as $a^{-9+9} = a^0$.

The Final Result

After following the order of operations and simplifying the expression, we arrive at the final result: $a^0$. But what does $a^0$ mean? In algebra, any non-zero number raised to the power of 0 is equal to 1. Therefore, $a^0 = 1$.

Conclusion

In this article, we have simplified the expression $\frac{a^3}{a^{12}} \times \left(a^3\right)^3$ using the rules of exponents and algebraic manipulation. We have followed the order of operations (PEMDAS) and arrived at the final result: $a^0 = 1$. This example demonstrates the importance of understanding the rules of exponents and algebraic manipulation in simplifying complex expressions.

Frequently Asked Questions

• What is the rule of exponents?: The rule of exponents states that when we multiply two numbers with the same base, we add their exponents. For example, $a^m \times a^n = a^{m+n}$. Similarly, when we divide two numbers with the same base, we subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.
• How do I simplify a complex expression?: To simplify a complex expression, follow the order of operations (PEMDAS): evaluate the exponentiation, simplify the fraction, and multiply the terms.
• What does $a^0$ mean?: In algebra, any non-zero number raised to the power of 0 is equal to 1. Therefore, $a^0 = 1$.

Additional Resources

• Algebraic Manipulation: This article provides a comprehensive guide to algebraic manipulation, including the rules of exponents and simplifying complex expressions.
• Rules of Exponents: This article provides a detailed explanation of the rules of exponents, including how to multiply and divide numbers with the same base.
• Simplifying Complex Expressions: This article provides a step-by-step guide to simplifying complex expressions using the rules of exponents and algebraic manipulation.
  

Introduction

In our previous article, we explored the world of algebraic manipulation and simplified the expression $\frac{a^3}{a^{12}} \times \left(a^3\right)^3$. We followed the order of operations (PEMDAS) and arrived at the final result: $a^0 = 1$. In this article, we will answer some frequently asked questions related to algebraic manipulation and simplifying complex expressions.

Q&A

Q: What is the rule of exponents?

A: The rule of exponents states that when we multiply two numbers with the same base, we add their exponents. For example, $a^m \times a^n = a^{m+n}$. Similarly, when we divide two numbers with the same base, we subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Q: How do I simplify a complex expression?

A: To simplify a complex expression, follow the order of operations (PEMDAS): evaluate the exponentiation, simplify the fraction, and multiply the terms.

Q: What does $a^0$ mean?

A: In algebra, any non-zero number raised to the power of 0 is equal to 1. Therefore, $a^0 = 1$.

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

A: Yes, you can simplify an expression with a negative exponent. For example, $a^{-3}$ can be rewritten as $\frac{1}{a^3}$.

Q: How do I handle exponents with different bases?

A: When you have exponents with different bases, you cannot simply add or subtract the exponents. Instead, you need to use the rule of exponents to rewrite the expression with a common base.

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

A: Yes, you can simplify an expression with a variable in the exponent. For example, $a^{2x}$ can be rewritten as $(a^2)^x$.

Q: How do I handle expressions with multiple variables?

A: When you have expressions with multiple variables, you need to use the rule of exponents to rewrite the expression with a common base. For example, $a^m \times b^n$ can be rewritten as $a^m \times b^n = a^m \times b^n$.

Q: Can I simplify an expression with a fraction in the exponent?

A: Yes, you can simplify an expression with a fraction in the exponent. For example, $a^{\frac{1}{2}}$ can be rewritten as $\sqrt{a}$.

Conclusion

In this article, we have answered some frequently asked questions related to algebraic manipulation and simplifying complex expressions. We have covered topics such as the rule of exponents, simplifying complex expressions, and handling exponents with different bases. By following the order of operations (PEMDAS) and using the rule of exponents, you can simplify even the most complex expressions.

Additional Resources

• Algebraic Manipulation: This article provides a comprehensive guide to algebraic manipulation, including the rules of exponents and simplifying complex expressions.
• Rules of Exponents: This article provides a detailed explanation of the rules of exponents, including how to multiply and divide numbers with the same base.
• Simplifying Complex Expressions: This article provides a step-by-step guide to simplifying complex expressions using the rules of exponents and algebraic manipulation.

Frequently Asked Questions

• What is the rule of exponents?: The rule of exponents states that when we multiply two numbers with the same base, we add their exponents. For example, $a^m \times a^n = a^{m+n}$. Similarly, when we divide two numbers with the same base, we subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.
• How do I simplify a complex expression?: To simplify a complex expression, follow the order of operations (PEMDAS): evaluate the exponentiation, simplify the fraction, and multiply the terms.
• What does $a^0$ mean?: In algebra, any non-zero number raised to the power of 0 is equal to 1. Therefore, $a^0 = 1$.

Tips and Tricks

• Use the rule of exponents to rewrite expressions with a common base: When you have expressions with different bases, use the rule of exponents to rewrite the expression with a common base.
• Follow the order of operations (PEMDAS): To simplify a complex expression, follow the order of operations (PEMDAS): evaluate the exponentiation, simplify the fraction, and multiply the terms.
• Use algebraic manipulation to simplify expressions: Algebraic manipulation is a powerful tool for simplifying complex expressions. By using the rules of exponents and algebraic manipulation, you can simplify even the most complex expressions.