Question

Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. $$ \frac{\left(16 x^{2} y^{-1 / 3}\right)^{3 / 4}}{\left(x y^{2}\right)^{1 / 4}} $$

Short answer

The simplified expression is \(\frac{8 x^{5/4}}{y^{3/4}}\).

Step-by-step solution

Simplify the numerator

Simplify the expression in the numerator \[(16 x^{2} y^{-1/3})^{3/4}\].Apply the power to each factor inside the parentheses:\[16^{3/4} (x^{2})^{3/4} (y^{-1/3})^{3/4}\].

Apply exponents

Calculate each term separately: \begin{align*}16^{3/4} &= (2^4)^{3/4} = 2^{4 \times 3/4} = 2^3 = 8, \ (x^{2})^{3/4} &= x^{2 \times 3/4} = x^{3/2}, \ (y^{-1/3})^{3/4} &= y^{-1/3 \times 3/4} = y^{-1/4}.dotleftTherefore, the numerator becomes \[8 x^{3/2} y^{-1/4}\].

Simplify the denominator

Simplify the expression in the denominator \[(x y^{2})^{1/4}\]. Apply the power to each factor inside the parentheses: \[(x^{1})^{1/4} (y^{2})^{1/4}\].

Apply exponents

Calculate each term separately: \[(x^{1})^{1/4} = x^{1/4}, \ (y^{2})^{1/4} = y^{1/2}\].Therefore, the denominator becomes \[x^{1/4} y^{1/2}\].

Combine and simplify

Combine the simplified numerator and denominator: \[\frac{8 x^{3/2} y^{-1/4}}{x^{1/4} y^{1/2}}\].Subtract the exponents of the same bases: \[8 x^{3/2 - 1/4} y^{-1/4 - 1/2} = 8 x^{6/4 - 1/4} y^{-1/4 - 2/4} = 8 x^{5/4} y^{-3/4}\].

Convert to positive exponents

Express the result using only positive exponents: \[8 x^{5/4} y^{-3/4} = 8 x^{5/4} \frac{1}{y^{3/4}} = \frac{8 x^{5/4}}{y^{3/4}}\].

Key concepts

positive exponents

Positive exponents indicate how many times to multiply the base by itself. For example, in the expression \(x^3\), the base is \(x\) and the exponent is 3, which means \(x \times x \times x\). Positive exponents simplify polynomial expressions and make computations easier to follow.
To convert negative exponents to positive exponents, move the term with the negative exponent to the opposite part of the fraction. For example, \( y^{-3/4} \) becomes \( \frac{1}{y^{3/4}} \).
This method helps in expressing your final results in a conventional and simplified form. Always aim to write your final answer with positive exponents to ensure clarity and standardization.

fractional exponents

Fractional exponents, also known as rational exponents, indicate both a root and a power. For example, \( x^{1/2} \) indicates the square root of \( x \), and \( x^{3/4} \) means the fourth root of \( x^3 \).
This concept is crucial in simplification processes because it allows a smooth transition between radical expressions and exponential expressions.
To simplify expressions with fractional exponents, remember the laws of exponents:

• \( (a^{m})^n = a^{m\cdot n} \)
• \( a^{m/n} = \sqrt[n]{a^m} \)

Using these properties, complex expressions become more manageable. It’s also pivotal in combining and reducing terms to their simplest forms.

combining exponents

Combining exponents follows certain rules to ensure the expression remains accurate and simplified. Here are some basic laws to remember:

• When multiplying like bases: \( a^m \cdot a^n = a^{m+n} \)
• When dividing like bases: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a power: \( (a^m)^n = a^{m \cdot n} \)

In our example, the final step involves subtracting the exponents of the same base in the numerator and denominator. This results in combining them properly:

• \( x^{3/2} - x^{1/4} = x^{6/4} - x^{1/4} = x^{5/4} \)
• \( y^{-1/4} - y^{1/2} = y^{-1/4} - y^{2/4} = y^{-3/4} \)

Following these steps ensures that your results are simplified accurately and consistently, providing a clear final answer.