UNDRAISERS The hospital spent $$ 9^{5} $$ dollars on new medical equipment this year. Last ear, they spent $$ 9^{7} $$ dollars. How many times more money did they spend last year than his year? IEASUREMENT 1 milligram is equal to $$ 10^{-3} $$ grams. Write this number using a positive xponent.

Question

UpStudy AI · Accepted Answer

Calculate the value by following steps:
- step0: Calculate:
  $$10^{-3}$$
- step1: Express with a positive exponent:
  $$\frac{1}{10^{3}}$$
- step2: Evaluate the power:
  $$\frac{1}{1000}$$
Calculate or simplify the expression $$ 9^7 / 9^5 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{9^{7}}{9^{5}}$$
- step1: Multiply by $$a^{-n}:$$
  $$9^{7}\times 9^{-5}$$
- step2: Multiply:
  $$9^{7-5}$$
- step3: Subtract the numbers:
  $$9^{2}$$
- step4: Evaluate the power:
  $$81$$
Let's break down the two parts of your problem step by step.

### Part 1: Hospital Spending Comparison

1. **Known Conditions**:
   - This year, the hospital spent $$ 9^{5} $$ dollars.
   - Last year, they spent $$ 9^{7} $$ dollars.

2. **Finding the Ratio**:
   To find out how many times more money was spent last year compared to this year, we can set up the ratio:
   $$
   \text{Ratio} = \frac{\text{Last Year Spending}}{\text{This Year Spending}} = \frac{9^{7}}{9^{5}}
   $$

3. **Simplifying the Expression**:
   Using the properties of exponents, we can simplify:
   $$
   \frac{9^{7}}{9^{5}} = 9^{7-5} = 9^{2}
   $$

4. **Calculating the Value**:
   Now, we calculate $$ 9^{2} $$:
   $$
   9^{2} = 81
   $$

Thus, the hospital spent **81 times more money last year than this year**.

### Part 2: Measurement Conversion

1. **Known Condition**:
   - 1 milligram is equal to $$ 10^{-3} $$ grams.

2. **Writing with a Positive Exponent**:
   To express $$ 10^{-3} $$ using a positive exponent, we can rewrite it as:
   $$
   10^{-3} = \frac{1}{10^{3}}
   $$

Thus, $$ 1 $$ milligram is equal to $$ \frac{1}{1000} $$ grams or $$ 0.001 $$ grams.

### Summary of Results:
- The hospital spent **81 times more** last year than this year.
- $$ 1 $$ milligram is equal to $$ \frac{1}{1000} $$ grams or $$ 0.001 $$ grams.
- The hospital spent 81 times more money last year than this year. - 1 milligram is equal to 0.001 grams.

UNDRAISERS The hospital spent \( 9^{5} \) dollars on new medical equipment this year. Last ear, they spent \( 9^{7} \) dollars. How many times more money did they spend last year than his year? IEASUREMENT 1 milligram is equal to \( 10^{-3} \) grams. Write this number using a positive xponent.

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