Simplify The Expression: 2 ⋅ 2 ⋅ 2 N + 2 − 2 N + 1 4 N + 1 2 \cdot 2 \cdot \frac{2^{n+2} - 2^{n+1}}{4^{n+1}} 2 ⋅ 2 ⋅ 4 N + 1 2 N + 2 − 2 N + 1 ​

Introduction

In this article, we will simplify the given expression: $2 \cdot 2 \cdot \frac{2^{n+2} - 2^{n+1}}{4^{n+1}}$. This expression involves exponents, multiplication, and division, making it a challenging problem to solve. We will break down the solution into manageable steps, using algebraic manipulations and properties of exponents to simplify the expression.

Step 1: Simplify the Numerator

The numerator of the expression is $2^{n+2} - 2^{n+1}$. We can simplify this expression by factoring out the common term $2^{n+1}$.

$$2^{n+2} - 2^{n+1} = 2^{n+1}(2 - 1)$$

Using the property of exponents that states $a^m \cdot a^n = a^{m+n}$, we can rewrite the expression as:

$$2^{n+1}(2 - 1) = 2^{n+1} \cdot 1$$

Simplifying further, we get:

$$2^{n+1} \cdot 1 = 2^{n+1}$$

Step 2: Simplify the Denominator

The denominator of the expression is $4^{n+1}$. We can simplify this expression by rewriting it in terms of base 2.

$$4^{n+1} = (2^2)^{n+1}$$

Using the property of exponents that states $(a^m)^n = a^{mn}$, we can rewrite the expression as:

$$(2^2)^{n+1} = 2^{2(n+1)}$$

Simplifying further, we get:

$$2^{2(n+1)} = 2^{2n+2}$$

Step 3: Simplify the Expression

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

$$2 \cdot 2 \cdot \frac{2^{n+1}}{2^{2n+2}}$$

Using the property of exponents that states $\frac{a^m}{a^n} = a^{m-n}$, we can rewrite the expression as:

$$2 \cdot 2 \cdot 2^{n+1-2n-2}$$

Simplifying further, we get:

$$2 \cdot 2 \cdot 2^{-n-1}$$

Using the property of exponents that states $a^m \cdot a^n = a^{m+n}$, we can rewrite the expression as:

$$2 \cdot 2 \cdot 2^{-n-1} = 2^{1-(-n-1)}$$

Simplifying further, we get:

$$2^{1-(-n-1)} = 2^{n+2}$$

Conclusion

In this article, we simplified the given expression: $2 \cdot 2 \cdot \frac{2^{n+2} - 2^{n+1}}{4^{n+1}}$. We broke down the solution into manageable steps, using algebraic manipulations and properties of exponents to simplify the expression. The final simplified expression is $2^{n+2}$.

Key Takeaways

• We can simplify the numerator of the expression by factoring out the common term $2^{n+1}$.
• We can simplify the denominator of the expression by rewriting it in terms of base 2.
• We can use the property of exponents that states $\frac{a^m}{a^n} = a^{m-n}$ to simplify the expression.
• We can use the property of exponents that states $a^m \cdot a^n = a^{m+n}$ to simplify the expression.

Frequently Asked Questions

• What is the simplified expression?
  • The simplified expression is $2^{n+2}$.
• How do we simplify the numerator of the expression?
  • We can simplify the numerator by factoring out the common term $2^{n+1}$.
• How do we simplify the denominator of the expression?
  • We can simplify the denominator by rewriting it in terms of base 2.

Further Reading

• Exponents and Powers: A Comprehensive Guide
• Algebraic Manipulations: A Step-by-Step Guide
• Properties of Exponents: A Summary
  Simplify the Expression: A Q&A Guide =====================================

Introduction

In our previous article, we simplified the given expression: $2 \cdot 2 \cdot \frac{2^{n+2} - 2^{n+1}}{4^{n+1}}$. We broke down the solution into manageable steps, using algebraic manipulations and properties of exponents to simplify the expression. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q&A

Q: What is the simplified expression?

A: The simplified expression is $2^{n+2}$.

Q: How do we simplify the numerator of the expression?

A: We can simplify the numerator by factoring out the common term $2^{n+1}$. This can be done by rewriting the expression as $2^{n+1}(2 - 1)$.

Q: How do we simplify the denominator of the expression?

A: We can simplify the denominator by rewriting it in terms of base 2. This can be done by rewriting the expression as $4^{n+1} = (2^2)^{n+1}$.

Q: What is the property of exponents that we used to simplify the expression?

A: We used the property of exponents that states $\frac{a^m}{a^n} = a^{m-n}$ to simplify the expression.

Q: What is the property of exponents that we used to simplify the expression further?

A: We used the property of exponents that states $a^m \cdot a^n = a^{m+n}$ to simplify the expression further.

Q: Can we simplify the expression further?

A: Yes, we can simplify the expression further by using the property of exponents that states $a^m \cdot a^n = a^{m+n}$. This can be done by rewriting the expression as $2^{1-(-n-1)}$.

Q: What is the final simplified expression?

A: The final simplified expression is $2^{n+2}$.

Common Mistakes

• Not factoring out the common term $2^{n+1}$ from the numerator.
• Not rewriting the denominator in terms of base 2.
• Not using the property of exponents that states $\frac{a^m}{a^n} = a^{m-n}$ to simplify the expression.
• Not using the property of exponents that states $a^m \cdot a^n = a^{m+n}$ to simplify the expression further.

Tips and Tricks

• Always factor out the common term from the numerator.
• Always rewrite the denominator in terms of base 2.
• Always use the property of exponents that states $\frac{a^m}{a^n} = a^{m-n}$ to simplify the expression.
• Always use the property of exponents that states $a^m \cdot a^n = a^{m+n}$ to simplify the expression further.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression: $2 \cdot 2 \cdot \frac{2^{n+2} - 2^{n+1}}{4^{n+1}}$. We provided tips and tricks to help you simplify the expression correctly. We also discussed common mistakes to avoid.

Further Reading

• Exponents and Powers: A Comprehensive Guide
• Algebraic Manipulations: A Step-by-Step Guide
• Properties of Exponents: A Summary

Resources

• Online calculators for simplifying expressions
• Algebra textbooks for further reading
• Online resources for practicing algebraic manipulations