Question

given \\(\frac{a^{5t}}{a^7}\\). for what value of \\(t\\) is the given expression equivalent to \\(a^{13}\\)? \\(t =\\)

Explanation:

Step1: Apply exponent division rule

When dividing like bases, subtract exponents: $\frac{a^{m}}{a^{n}} = a^{m-n}$. So, $\frac{a^{5t}}{a^{7}} = a^{5t - 7}$.

Step2: Set exponents equal

Since the expression equals $a^{13}$, set the exponents equal: $5t - 7 = 13$.

Step3: Solve for t (add 7 to both sides)

$5t = 13 + 7$
$5t = 20$

Step4: Isolate t (divide by 5)

$t = \frac{20}{5}$

Answer:

$t = 4$