Simplify The Expression:${ \frac{9 2}{3 4} - 9^0 }$

Introduction

In this article, we will simplify the given expression $\frac{9^2}{3^4} - 9^0$. This involves applying the rules of exponents and simplifying the resulting expression. We will break down the problem step by step, using the properties of exponents to simplify the expression.

Understanding Exponents

Before we start simplifying the expression, let's review the rules of exponents. The exponent of a number is the power to which the number is raised. For example, in the expression $3^4$, the exponent is 4, and the base is 3. The value of the expression is equal to the base raised to the power of the exponent, which is $3^4 = 3 \times 3 \times 3 \times 3 = 81$.

Simplifying the Expression

Now that we have reviewed the rules of exponents, let's simplify the given expression. The expression is $\frac{9^2}{3^4} - 9^0$. We can start by simplifying the numerator and denominator separately.

Simplifying the Numerator

The numerator of the expression is $9^2$. We can simplify this by applying the rule of exponents that states $a^m \times a^n = a^{m+n}$. In this case, we have $9^2 = (3^2)^2 = 3^4$.

Simplifying the Denominator

The denominator of the expression is $3^4$. This is already simplified, so we can move on to the next step.

Simplifying the Expression

Now that we have simplified the numerator and denominator, we can rewrite the expression as $\frac{3^4}{3^4} - 9^0$. We can simplify this by applying the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$, where $m$ and $n$ are exponents. In this case, we have $\frac{3^4}{3^4} = 3^{4-4} = 3^0$.

Simplifying the Final Expression

The final expression is $3^0 - 9^0$. We can simplify this by applying the rule of exponents that states $a^0 = 1$, where $a$ is any non-zero number. In this case, we have $3^0 = 1$ and $9^0 = 1$. Therefore, the final expression is $1 - 1 = 0$.

Conclusion

In this article, we simplified the expression $\frac{9^2}{3^4} - 9^0$ by applying the rules of exponents. We broke down the problem step by step, simplifying the numerator and denominator separately. We then applied the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$ to simplify the expression. Finally, we applied the rule of exponents that states $a^0 = 1$ to simplify the final expression. The result is $0$.

Frequently Asked Questions

• What is the rule of exponents that states $a^m \times a^n = a^{m+n}$? The rule of exponents that states $a^m \times a^n = a^{m+n}$ is a fundamental rule in mathematics that allows us to simplify expressions involving exponents.
• What is the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$? The rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$ is a fundamental rule in mathematics that allows us to simplify expressions involving exponents.
• What is the rule of exponents that states $a^0 = 1$? The rule of exponents that states $a^0 = 1$ is a fundamental rule in mathematics that allows us to simplify expressions involving exponents.

Final Answer

The final answer is $\boxed{0}$.

Introduction

In our previous article, we simplified the expression $\frac{9^2}{3^4} - 9^0$ by applying the rules of exponents. We broke down the problem step by step, simplifying the numerator and denominator separately. We then applied the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$ to simplify the expression. Finally, we applied the rule of exponents that states $a^0 = 1$ to simplify the final expression. The result is $0$.

Q&A

Q: What is the rule of exponents that states $a^m \times a^n = a^{m+n}$?

A: The rule of exponents that states $a^m \times a^n = a^{m+n}$ is a fundamental rule in mathematics that allows us to simplify expressions involving exponents. This rule states that when we multiply two numbers with the same base, we can add their exponents.

Q: What is the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$?

A: The rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$ is a fundamental rule in mathematics that allows us to simplify expressions involving exponents. This rule states that when we divide two numbers with the same base, we can subtract their exponents.

Q: What is the rule of exponents that states $a^0 = 1$?

A: The rule of exponents that states $a^0 = 1$ is a fundamental rule in mathematics that allows us to simplify expressions involving exponents. This rule states that any non-zero number raised to the power of zero is equal to 1.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you can follow these steps:

1. Simplify the numerator and denominator separately.
2. Apply the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$ to simplify the expression.
3. Apply the rule of exponents that states $a^0 = 1$ to simplify the final expression.

Q: What is the difference between a base and an exponent?

A: A base is the number that is being raised to a power, while an exponent is the power to which the base is being raised. For example, in the expression $3^4$, the base is 3 and the exponent is 4.

Q: Can I simplify an expression with negative exponents?

A: Yes, you can simplify an expression with negative exponents by applying the rule of exponents that states $a^{-n} = \frac{1}{a^n}$. This rule states that a negative exponent is equal to the reciprocal of the positive exponent.

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, you can follow these steps:

1. Simplify the numerator and denominator separately.
2. Apply the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$ to simplify the expression.
3. Apply the rule of exponents that states $a^0 = 1$ to simplify the final expression.

Conclusion

In this article, we answered some of the most frequently asked questions about simplifying expressions with exponents. We covered topics such as the rule of exponents that states $a^m \times a^n = a^{m+n}$, the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$, and the rule of exponents that states $a^0 = 1$. We also provided tips and tricks for simplifying expressions with exponents, including how to simplify expressions with negative exponents and multiple exponents.

Final Answer

The final answer is $\boxed{0}$.