Question

Complete the over/under table to estimate the solution to 8^x = 160

1. 8^2 =64 : _
2. 8^3 = 512 : _
3. 8^2.5 = 181.02 : _
4. 8^2.4 = 147.03 : _
5. 8^2.45 = 163.14 : _
6. 8^2.44 = 159.79 : _

Apply Over / under to every equation

A. Over
B. Under

Use the over/under table to estimate the solution to the equation to the nearest hundredth

8^x = 160

A. Over
B. Under

Answer 1

To estimate the solution to the equation \( 8^x = 160 \), let's assess each of the provided equations using over/under logic:

1. \( 8^2 = 64 \): Under (because \( 64 < 160 \))
2. \( 8^3 = 512 \): Over (because \( 512 > 160 \))
3. \( 8^{2.5} = 181.02 \): Over (because \( 181.02 > 160 \))
4. \( 8^{2.4} = 147.03 \): Under (because \( 147.03 < 160 \))
5. \( 8^{2.45} = 163.14 \): Over (because \( 163.14 > 160 \))
6. \( 8^{2.44} = 159.79 \): Under (because \( 159.79 < 160 \))

Now we can summarize the over/under status as follows:

- \( 8^2 \): Under
- \( 8^3 \): Over
- \( 8^{2.5} \): Over
- \( 8^{2.4} \): Under
- \( 8^{2.45} \): Over
- \( 8^{2.44} \): Under

Now let's apply the over/under logic to estimate \( x \) in \( 8^x = 160 \):

- Since \( 8^{2.4} < 160 < 8^{2.5} \), this leads us to anticipate that \( x \) must be between \( 2.4 \) and \( 2.5 \).
- Between \( 2.44 \) (Under) and \( 2.45 \) (Over), we anticipate that \( x \) is slightly more than \( 2.44 \) and less than \( 2.45 \).

Therefore, we can conclude:

**Estimate for \( 8^x = 160 \)**:

The value of \( x \) is approximately \( 2.44 \) to \( 2.45 \).

To summarize:

- The nearest estimate for \( x \) would be around **2.44** to **2.45.**

Thus, applying the over/under logic, the final answer is:

A. Over (for \( 2.45 \) since \( 8^{2.45} > 160 \))
B. Under (for \( 2.44 \) since \( 8^{2.44} < 160 \))

So realistically, the estimate is around \( 2.44 \) or just above.