Simplify (4^(2+n)*2^(4n+1)*8^(2-n))/(2^(3n+1))
The given problem is an algebraic expression that involves simplifying a complex fraction composed of exponential terms. The numerator of the fraction contains three exponential expressions with bases of 4, 2, and 8 raised to various powers involving 'n', which is a variable. These terms are multiplied together. The denominator contains a single exponential expression with a base of 2, raised to a power that also involves the variable 'n'. The task is to apply the rules of exponents to simplify the entire expression, which would involve combining like terms, using exponent addition, subtraction, and perhaps division rules to simplify the exponentials to the simplest form possible.
$\frac{4^{2 + n} \cdot 2^{4 n + 1} \cdot 8^{2 - n}}{2^{3 n + 1}}$
Answer
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:
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).
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.
Distributive Property: This property allows you to multiply a sum by a factor by multiplying each addend separately and then sum the products.
Simplifying Exponents: When you have an expression with exponents, you can often simplify it by combining like terms or applying exponent rules.
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.
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.
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 $1$.
