Write in simplified radical form with at most one radical.
$\frac{\sqrt{b c}}{\sqrt[3]{b c}}$
Assume that the variables represent positive real numbers.
Answer (1)
∙ Rewrite the radicals as fractional exponents: b c = ( b c ) 2 1 and 3 b c = ( b c ) 3 1 .
∙ Apply the quotient rule for exponents: ( b c ) 3 1 ( b c ) 2 1 = ( b c ) 2 1 − 3 1 .
∙ Calculate the difference of the fractions: 2 1 − 3 1 = 6 1 .
∙ Rewrite the fractional exponent as a radical: ( b c ) 6 1 = 6 b c . The final answer is 6 b c .
Explanation
Understanding the Problem We are asked to simplify the expression 3 b c b c into a single radical, assuming that b and c are positive real numbers.
Converting to Fractional Exponents First, we rewrite the radicals using fractional exponents: b c = ( b c ) 2 1 3 b c = ( b c ) 3 1
Rewriting the Expression Now, we rewrite the original expression using these fractional exponents: 3 b c b c = ( b c ) 3 1 ( b c ) 2 1
Applying the Quotient Rule Using the quotient rule for exponents, which states that x n x m = x m − n , we have: ( b c ) 3 1 ( b c ) 2 1 = ( b c ) 2 1 − 3 1
Calculating the Exponent We need to find the difference between the fractions 2 1 and 3 1 . To do this, we find a common denominator, which is 6: 2 1 − 3 1 = 6 3 − 6 2 = 6 1
Simplifying the Exponent So, our expression becomes: ( b c ) 6 1
Converting Back to Radical Form Finally, we convert the fractional exponent back to a radical: ( b c ) 6 1 = 6 b c Therefore, the simplified expression is 6 b c .
Examples
Imagine you are comparing the side lengths of two different containers. One container's side length is expressed as the square root of a product, and another's is the cube root of the same product. Simplifying the ratio of these side lengths, as we did in the problem, allows you to easily compare their sizes and understand their relationship. This type of simplification is useful in various fields, such as engineering, where comparing dimensions and scaling factors is essential for design and analysis.
