Write an exponential function in the form $$ y=a b^{x} $$ that goes through the points $$ (0,20) $$ and $$ (3,2500) $$.

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To find an exponential function of the form $$ y = a \cdot b^{x} $$ that passes through the points $$ (0, 20) $$ and $$ (3, 2500) $$, follow these steps:

1. **Determine the value of $$ a $$:**

When $$ x = 0 $$:
   $$
   y = a \cdot b^{0} = a \cdot 1 = a
   $$
   Given that $$ y = 20 $$ when $$ x = 0 $$, we have:
   $$
   a = 20
   $$

2. **Determine the value of $$ b $$:**

Using the point $$ (3, 2500) $$:
   $$
   2500 = 20 \cdot b^{3}
   $$
   Solve for $$ b^{3} $$:
   $$
   b^{3} = \frac{2500}{20} = 125
   $$
   Take the cube root of both sides:
   $$
   b = \sqrt[3]{125} = 5
   $$

3. **Write the exponential function:**

Substituting $$ a = 20 $$ and $$ b = 5 $$ into the general form:
   $$
   y = 20 \cdot 5^{x}
   $$

**Final Answer:**
$$
y = 20 \times 5^{\,x}
$$
The exponential function is $$ y = 20 \times 5^{x} $$.

Write an exponential function in the form \( y=a b^{x} \) that goes through the points \( (0,20) \) and \( (3,2500) \).

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