Simplify The Expression:${ \frac{\left(2 3\right) 0}{2^3 \cdot\left(2 4\right) 2} }$

Introduction

When it comes to simplifying expressions involving exponents and fractions, it's essential to understand the rules and properties of exponents. In this article, we will focus on simplifying the given expression: $\frac{\left(2^3\right)^0}{2^3 \cdot\left(2^4\right)^2}$. We will break down the expression into smaller parts, apply the rules of exponents, and simplify the resulting fraction.

Understanding Exponents

Before we dive into simplifying the expression, let's review the basics of exponents. An exponent is a small number that is written to the right of a base number and indicates how many times the base number should be multiplied by itself. For example, $2^3$ means $2$ multiplied by itself $3$ times, or $2 \times 2 \times 2 = 8$. The exponent $3$ is called the power or the index.

Simplifying the Expression

Now that we have a basic understanding of exponents, let's simplify the given expression. The expression is $\frac{\left(2^3\right)^0}{2^3 \cdot\left(2^4\right)^2}$. To simplify this expression, we need to apply the rules of exponents.

Rule 1: Zero Exponent

The first rule we need to apply is the zero exponent rule, which states that any number raised to the power of $0$ is equal to $1$. In this case, we have $\left(2^3\right)^0$, which simplifies to $1$.

Rule 2: Product of Powers

The second rule we need to apply is the product of powers rule, which states that when we multiply two numbers with the same base, we add their exponents. In this case, we have $2^3 \cdot\left(2^4\right)^2$, which simplifies to $2^{3+8} = 2^{11}$.

Rule 3: Quotient of Powers

The third rule we need to apply is the quotient of powers rule, which states that when we divide two numbers with the same base, we subtract their exponents. In this case, we have $\frac{\left(2^3\right)^0}{2^3 \cdot\left(2^4\right)^2}$, which simplifies to $\frac{1}{2^{11}}$.

Simplifying the Fraction

Now that we have simplified the expression, we need to simplify the resulting fraction. A fraction is simplified when the numerator and denominator have no common factors. In this case, the numerator is $1$ and the denominator is $2^{11}$. Since $1$ and $2^{11}$ have no common factors, the fraction is already simplified.

Conclusion

In conclusion, we have simplified the given expression $\frac{\left(2^3\right)^0}{2^3 \cdot\left(2^4\right)^2}$ using the rules of exponents. We applied the zero exponent rule, the product of powers rule, and the quotient of powers rule to simplify the expression. The resulting simplified fraction is $\frac{1}{2^{11}}$.

Frequently Asked Questions

• What is the zero exponent rule? The zero exponent rule states that any number raised to the power of $0$ is equal to $1$.
• What is the product of powers rule? The product of powers rule states that when we multiply two numbers with the same base, we add their exponents.
• What is the quotient of powers rule? The quotient of powers rule states that when we divide two numbers with the same base, we subtract their exponents.

Final Answer

The final answer is $\boxed{\frac{1}{2048}}$.

Introduction

In our previous article, we simplified the expression $\frac{\left(2^3\right)^0}{2^3 \cdot\left(2^4\right)^2}$ using the rules of exponents. We applied the zero exponent rule, the product of powers rule, and the quotient of powers rule to simplify the expression. The resulting simplified fraction is $\frac{1}{2^{11}}$. In this article, we will answer some frequently asked questions related to simplifying expressions involving exponents and fractions.

Q&A

Q: What is the zero exponent rule?

A: The zero exponent rule states that any number raised to the power of $0$ is equal to $1$. For example, $2^0 = 1$.

Q: What is the product of powers rule?

A: The product of powers rule states that when we multiply two numbers with the same base, we add their exponents. For example, $2^3 \cdot 2^4 = 2^{3+4} = 2^7$.

Q: What is the quotient of powers rule?

A: The quotient of powers rule states that when we divide two numbers with the same base, we subtract their exponents. For example, $\frac{2^3}{2^4} = 2^{3-4} = 2^{-1}$.

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

A: To simplify an expression with multiple exponents, we need to apply the rules of exponents. We can start by simplifying the exponents using the zero exponent rule, the product of powers rule, and the quotient of powers rule. Then, we can simplify the resulting expression using the rules of fractions.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the base number should be multiplied by itself a certain number of times. For example, $2^3$ means $2$ multiplied by itself $3$ times. A negative exponent indicates that the base number should be divided by itself a certain number of times. For example, $2^{-3}$ means $1$ divided by $2$ multiplied by itself $3$ times.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, we need to apply the quotient of powers rule. We can rewrite the expression with a positive exponent by taking the reciprocal of the base number. For example, $\frac{1}{2^{-3}} = 2^3$.

Q: What is the difference between a fraction and a decimal?

A: A fraction is a way of expressing a part of a whole as a ratio of two numbers. For example, $\frac{1}{2}$ means one half. A decimal is a way of expressing a fraction as a number with a point separating the whole number part from the fractional part. For example, $0.5$ means one half.

Q: How do I convert a fraction to a decimal?

A: To convert a fraction to a decimal, we need to divide the numerator by the denominator. For example, $\frac{1}{2} = 0.5$.

Conclusion

In conclusion, we have answered some frequently asked questions related to simplifying expressions involving exponents and fractions. We have covered the zero exponent rule, the product of powers rule, and the quotient of powers rule. We have also discussed how to simplify expressions with multiple exponents, positive exponents, and negative exponents. Finally, we have covered the difference between a fraction and a decimal, and how to convert a fraction to a decimal.

Final Answer

The final answer is $\boxed{\frac{1}{2048}}$.