Question

use the properties of exponents to multiply.

1. ( x^{\frac{1}{3}} cdot x^{\frac{1}{6}} cdot x^{\frac{1}{4}} )
2. ( a^{\frac{2}{5}} cdot a^{\frac{3}{10}} cdot a^{\frac{2}{15}} )
3. ( m^{\frac{4}{7}} cdot m^{\frac{3}{14}} cdot m^{\frac{5}{28}} )

Explanation:

Problem 15: \( x^{\frac{1}{3}} \cdot x^{\frac{1}{6}} \cdot x^{\frac{1}{4}} \)

Step1: Apply exponent product rule (\(a^m \cdot a^n = a^{m + n}\))

\( x^{\frac{1}{3} + \frac{1}{6} + \frac{1}{4}} \)

Step2: Find common denominator (12) and add fractions

\( \frac{1}{3} = \frac{4}{12}, \frac{1}{6} = \frac{2}{12}, \frac{1}{4} = \frac{3}{12} \)
\( \frac{4}{12} + \frac{2}{12} + \frac{3}{12} = \frac{9}{12} = \frac{3}{4} \)

Step3: Simplify the exponent

\( x^{\frac{3}{4}} \)

Step1: Apply exponent product rule

\( a^{\frac{2}{5} + \frac{3}{10} + \frac{2}{15}} \)

Step2: Find common denominator (30) and add fractions

\( \frac{2}{5} = \frac{12}{30}, \frac{3}{10} = \frac{9}{30}, \frac{2}{15} = \frac{4}{30} \)
\( \frac{12}{30} + \frac{9}{30} + \frac{4}{30} = \frac{25}{30} = \frac{5}{6} \)

Step3: Simplify the exponent

\( a^{\frac{5}{6}} \)

Step1: Apply exponent product rule

\( m^{\frac{4}{7} + \frac{3}{14} + \frac{5}{28}} \)

Step2: Find common denominator (28) and add fractions

\( \frac{4}{7} = \frac{16}{28}, \frac{3}{14} = \frac{6}{28}, \frac{5}{28} = \frac{5}{28} \)
\( \frac{16}{28} + \frac{6}{28} + \frac{5}{28} = \frac{27}{28} \)

Step3: Simplify the exponent

\( m^{\frac{27}{28}} \)

Answer:

\( x^{\frac{3}{4}} \)

Problem 16: \( a^{\frac{2}{5}} \cdot a^{\frac{3}{10}} \cdot a^{\frac{2}{15}} \)