Simplify (4^(2+n)*2^(4n+1)*8^(2-n))/(2^(3n+1))

Solution:

Step:1

Eliminate the common base of $2^{4n+1}$ and $2^{3n+1}$.

Step:1.1

Extract $2^{3n+1}$ from $4^{2+n}\cdot 2^{4n+1}\cdot 8^{2-n}$.

$$\frac{2^{3n+1}(4^{2+n}\cdot 2^{4n+1-(3n+1)}\cdot 8^{2-n})}{2^{3n+1}}$$

Step:1.2

Remove the common bases.

Step:1.2.1

Introduce a multiplicative identity of $1$.

$$\frac{2^{3n+1}(4^{2+n}\cdot 2^{4n+1-(3n+1)}\cdot 8^{2-n})}{2^{3n+1}\cdot 1}$$

Step:1.2.2

Eliminate the shared base.

$$\frac{\cancel{2^{3n+1}}(4^{2+n}\cdot 2^{4n+1-(3n+1)}\cdot 8^{2-n})}{\cancel{2^{3n+1}}\cdot 1}$$

Step:1.2.3

Reformulate the expression.

$$\frac{4^{2+n}\cdot 2^{4n+1-(3n+1)}\cdot 8^{2-n}}{1}$$

Step:1.2.4

Divide the expression by $1$.

$$4^{2+n}\cdot 2^{4n+1-(3n+1)}\cdot 8^{2-n}$$

Step:2

Break down each term.

Step:2.1

Apply the distributive property.

$$4^{2+n}\cdot 2^{(4n+1)-(3n+1)}\cdot 8^{2-n}$$

Step:2.2

Simplify the exponent by subtraction.

$$4^{2+n}\cdot 2^n\cdot 8^{2-n}$$

Step:3

Combine like terms.

Step:3.1

Add the exponents of the powers of $2$.

$$2^{4(2+n)}\cdot 2^n\cdot 2^{3(2-n)}$$

Step:3.2

Use the exponent addition rule.

$$2^{8+4n+n+6-3n}$$

Step:3.3

Combine the terms.

$$2^{14+n}$$

Step:4

Evaluate the power of $2$.

Step:4.1

Recognize that $2^{14}$ is a constant.

$$2^{14}\cdot 2^n$$

Step:4.2

Calculate $2^{14}$.

$$16384\cdot 2^n$$

Step:5

Final simplification.

Step:5.1

The expression simplifies to:

$$16384\cdot 2^n$$

Knowledge Notes:

1. Common Bases: When simplifying expressions with exponents, if the bases are the same, you can add or subtract the exponents depending on the operation (multiplication or division).

2. Exponent Rules: The power rule $(a^m{)}^n=a^{mn}$ allows you to multiply exponents when a base is raised to a power, and then that result is raised to another power. The product rule $a^m\cdot a^n=a^{m+n}$ allows you to add exponents when multiplying like bases.

3. Distributive Property: This property allows you to multiply a sum by a factor by multiplying each addend separately and then sum the products.

4. Simplifying Exponents: When you have an expression with exponents, you can often simplify it by combining like terms or applying exponent rules.

5. Powers of Two: Knowing the powers of two (e.g., $2^1=2$, $2^2=4$, $2^3=8$, etc.) can be very helpful in simplifying expressions quickly without a calculator.

6. Multiplicative Identity: The number $1$ is the multiplicative identity because any number multiplied by $1$ remains unchanged. This is often used to simplify expressions where a term is divided by itself.

7. Cancellation: When the same term appears in both the numerator and the denominator, it can be canceled out, which is essentially dividing the term by itself to get $1$.