Question

calculate the value of the fraction \\(\dfrac{13^{\frac{3}{7}}}{13^{\frac{5}{7}}}\\)

Explanation:

Step1: Recall the exponent rule for division

When dividing two numbers with the same base, we subtract the exponents: $a^m \div a^n = a^{m - n}$. Here, we can rewrite the given expression as a division of two exponents with base 13: $13^{\frac{3}{7}} \div 13^{\frac{5}{7}}$.

Step2: Apply the exponent division rule

Using the rule $a^m \div a^n = a^{m - n}$, where $a = 13$, $m=\frac{3}{7}$, and $n = \frac{5}{7}$, we get $13^{\frac{3}{7}-\frac{5}{7}}$.

Step3: Simplify the exponent

Calculate the exponent: $\frac{3}{7}-\frac{5}{7}=\frac{3 - 5}{7}=\frac{-2}{7}$. So the expression becomes $13^{-\frac{2}{7}}$.

Step4: Rewrite the negative exponent

A negative exponent means the reciprocal of the positive exponent: $a^{-n}=\frac{1}{a^n}$. So $13^{-\frac{2}{7}}=\frac{1}{13^{\frac{2}{7}}}$.

Step5: Rewrite the fractional exponent as a root

Recall that $a^{\frac{m}{n}}=\sqrt[n]{a^m}$. So $13^{\frac{2}{7}}=\sqrt[7]{13^2}=\sqrt[7]{169}$. Thus, the expression is $\frac{1}{\sqrt[7]{169}}$ or we can also write it as $169^{-\frac{1}{7}}$.

Answer:

$13^{-\frac{2}{7}}$ (or $\frac{1}{\sqrt[7]{169}}$ or $169^{-\frac{1}{7}}$)