Question
Multiply \( 5^{\frac{1}{7}} \cdot 5^{\frac{3}{14}} \)
Ask by Lewis Campbell. in the United States
Jan 21,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( 5^{\frac{5}{14}} \)
Solution
To multiply \( 5^{\frac{1}{7}} \) and \( 5^{\frac{3}{14}} \), you can use the property of exponents that states:
\[
a^m \cdot a^n = a^{m + n}
\]
Applying this property:
\[
5^{\frac{1}{7}} \cdot 5^{\frac{3}{14}} = 5^{\frac{1}{7} + \frac{3}{14}}
\]
First, find a common denominator to add the exponents:
\[
\frac{1}{7} = \frac{2}{14}, \quad \text{so} \quad \frac{2}{14} + \frac{3}{14} = \frac{5}{14}
\]
Therefore:
\[
5^{\frac{1}{7}} \cdot 5^{\frac{3}{14}} = 5^{\frac{5}{14}}
\]
**Final Answer:** \( 5^{\,5/14} \)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To multiply the expressions \( 5^{\frac{1}{7}} \cdot 5^{\frac{3}{14}} \), we can use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \). First, we need to add the exponents: \[ \frac{1}{7} + \frac{3}{14} \] To add these fractions, we need a common denominator. The least common multiple of 7 and 14 is 14. We convert \( \frac{1}{7} \) into a fraction with a denominator of 14: \[ \frac{1}{7} = \frac{2}{14} \] Now we can add the two fractions: \[ \frac{2}{14} + \frac{3}{14} = \frac{2 + 3}{14} = \frac{5}{14} \] Now we can combine the expressions: \[ 5^{\frac{1}{7}} \cdot 5^{\frac{3}{14}} = 5^{\frac{5}{14}} \] Thus, the final result is: \[ 5^{\frac{5}{14}} \]
