Question

If you know that the product of two powers is 5 to the power of 3, and that the quotient is 5 to the power of 17, what could the two powers be? Explain how you know this answer is correct.

Answer 1

Answer:

$5^{10} \cdot 5^{-7} = 5^3$

$\frac{5^{10}}{5^{-7}} = 5^{17}$

Step-by-step explanation:

You have two equations:

$5^a \cdot 5^b = 5^3$

$\frac{5^a}{5^b} = 5^{17}$

But since

$x^a \cdot x^b = x^{a+b}$

you only need to solve a+b = 3 and a-b=17

Rewrite the second as a = 17+b and plug it into the first:

17+b + b = 3, then you find, 2b = -14, so b = -7.

Then a=3-b, so a = 10.