Simplify (((x^2y)^4)^(1/2))/(((x^2y)^(1/2))^4)
This problem involves simplifying an algebraic expression that uses exponents and involves a quotient of two large expressions. The expression in the numerator is a power raised to the fourth power and then to the one-half power. The expression in the denominator is the same base raised to the one-half power, then taken to the fourth power. The base for both the numerator and the denominator is the same: (x^2y). The question asks to simplify this complex expression by using the properties of exponents, such as the power of a power rule, the product to a power rule, and the quotient of powers rule.
$\frac{\left(\left(\right. \left(\left(\right. x^{2} y \left.\right)\right)^{4} \left.\right)\right)^{\frac{1}{2}}}{\left(\left(\right. \left(\left(\right. x^{2} y \left.\right)\right)^{\frac{1}{2}} \left.\right)\right)^{4}}$
Answer
Solution:
Step:1
Rework the numerator.
Step:1.1
Raise the power inside the parentheses in $\left(x^{2}y\right)^{4}$ to the power of $\frac{1}{2}$.
Step:1.1.1
Utilize the rule of exponents to raise a power to a power, $\left(a^{m}\right)^{n} = a^{mn}$, to get $\frac{\left(x^{2}y\right)^{4 \cdot \frac{1}{2}}}{\left(\left(x^{2}y\right)^{\frac{1}{2}}\right)^{4}}$.
Step:1.1.2
Simplify by reducing the exponent.
Step:1.1.2.1
Extract the factor of $2$ from $4$ to obtain $\frac{\left(x^{2}y\right)^{2 \cdot 2 \cdot \frac{1}{2}}}{\left(\left(x^{2}y\right)^{\frac{1}{2}}\right)^{4}}$.
Step:1.1.2.2
Eliminate the common factor to get $\frac{\left(x^{2}y\right)^{\cancel{2} \cdot 2 \cdot \frac{1}{\cancel{2}}}}{\left(\left(x^{2}y\right)^{\frac{1}{2}}\right)^{4}}$.
Step:1.1.2.3
Reformulate the expression as $\frac{\left(x^{2}y\right)^{2}}{\left(\left(x^{2}y\right)^{\frac{1}{2}}\right)^{4}}$.
Step:1.2
Apply the distributive property to $x^{2}y$ to get $\frac{\left(x^{2}\right)^{2}y^{2}}{\left(\left(x^{2}y\right)^{\frac{1}{2}}\right)^{4}}$.
Step:1.3
Raise the power inside the parentheses in $\left(x^{2}\right)^{2}$.
Step:1.3.1
Use the rule of exponents to raise a power to a power, $\left(a^{m}\right)^{n} = a^{mn}$, resulting in $\frac{x^{2 \cdot 2}y^{2}}{\left(\left(x^{2}y\right)^{\frac{1}{2}}\right)^{4}}$.
Step:1.3.2
Calculate $2$ times $2$ to simplify the expression to $\frac{x^{4}y^{2}}{\left(\left(x^{2}y\right)^{\frac{1}{2}}\right)^{4}}$.
Step:2
Streamline the denominator.
Step:2.1
Raise the power inside the parentheses in $\left(\left(x^{2}y\right)^{\frac{1}{2}}\right)^{4}$.
Step:2.1.1
Invoke the rule of exponents to raise a power to a power, $\left(a^{m}\right)^{n} = a^{mn}$, to get $\frac{x^{4}y^{2}}{\left(x^{2}y\right)^{\frac{1}{2} \cdot 4}}$.
Step:2.1.2
Reduce the exponent by simplifying.
Step:2.1.2.1
Extract the factor of $2$ from $4$ to obtain $\frac{x^{4}y^{2}}{\left(x^{2}y\right)^{\frac{1}{2} \cdot \left(2 \cdot 2\right)}}$.
Step:2.1.2.2
Eliminate the common factor to get $\frac{x^{4}y^{2}}{\left(x^{2}y\right)^{\frac{1}{\cancel{2}} \cdot \left(\cancel{2} \cdot 2\right)}}$.
Step:2.1.2.3
Reformulate the expression as $\frac{x^{4}y^{2}}{\left(x^{2}y\right)^{2}}$.
Step:2.2
Apply the distributive property to $x^{2}y$ to get $\frac{x^{4}y^{2}}{\left(x^{2}\right)^{2}y^{2}}$.
Step:2.3
Raise the power inside the parentheses in $\left(x^{2}\right)^{2}$.
Step:2.3.1
Utilize the rule of exponents to raise a power to a power, $\left(a^{m}\right)^{n} = a^{mn}$, resulting in $\frac{x^{4}y^{2}}{x^{2 \cdot 2}y^{2}}$.
Step:2.3.2
Calculate $2$ times $2$ to simplify the expression to $\frac{x^{4}y^{2}}{x^{4}y^{2}}$.
Step:3
Condense the expression by cancelling out identical factors.
Step:3.1
Eliminate the common $x^{4}$ factor.
Step:3.1.1
Cancel out the common factor to get $\frac{\cancel{x^{4}}y^{2}}{\cancel{x^{4}}y^{2}}$.
Step:3.1.2
Reformulate the expression as $\frac{y^{2}}{y^{2}}$.
Step:3.2
Eliminate the common $y^{2}$ factor.
Step:3.2.1
Cancel out the common factor to get $\frac{\cancel{y^{2}}}{\cancel{y^{2}}}$.
Step:3.2.2
Reformulate the expression as $1$.
Knowledge Notes:
The problem involves simplifying an expression with exponents by applying the rules of exponents. The relevant knowledge points include:
Power Rule: When raising an exponent to another exponent, you multiply the exponents together: $\left(a^{m}\right)^{n} = a^{mn}$.
Product Rule: When multiplying two powers with the same base, you add the exponents: $a^{m} \cdot a^{n} = a^{m+n}$.
Quotient Rule: When dividing two powers with the same base, you subtract the exponents: $\frac{a^{m}}{a^{n}} = a^{m-n}$.
Simplifying Expressions: When you have the same factor in the numerator and denominator, you can cancel them out.
Exponent of 1: Any number raised to the power of 1 is itself: $a^{1} = a$.
Exponent of 0: Any nonzero number raised to the power of 0 is 1: $a^{0} = 1$ (assuming $a \neq 0$).
In this problem, we applied the power rule to simplify the expression inside the parentheses and then used the quotient rule to divide the powers with the same base. Finally, we canceled out the common factors to arrive at the simplest form of the expression.
