Which of the following has the highest value?
(a)10 power 11
(b)12 power 9
(c)5 power 5
(d)11 power 10

Options

A. 10 power 11
B. 12 power 9
C. 5 power 5
D. 11 power 10

Answer

To determine which of the following options has the highest value, we will evaluate each option step by step.

**Step 1: Evaluate each option.**

- **Option (a):** \(10^{11}\)

- **Option (b):** \(12^{9}\)

- **Option (c):** \(5^{5}\)

- **Option (d):** \(11^{10}\)

**Step 2: Calculate the approximate values of each option.**

1. **Calculating \(10^{11}\):**
\[ 10^{11} = 100000000000 \quad \text{(which is 1 followed by 11 zeros)} \]

2. **Calculating \(12^{9}\):**
To find \(12^{9}\), we can break it down:
\[ 12^{9} = (12^3)^3 = 1728^3 \]
(Calculating \(12^3\) gives us \(1728\), which is a large number, and cubing it will yield a very large value.)

3. **Calculating \(5^{5}\):**
\[ 5^{5} = 3125 \quad \text{(which is relatively small compared to the others)} \]

4. **Calculating \(11^{10}\):**
To find \(11^{10}\), we can break it down:
\[ 11^{10} = (11^5)^2 \]
First, calculate \(11^5\):
\[ 11^5 = 161051 \quad \text{(which is already a large number)} \]
Then square it:
\[ 11^{10} = 161051^2 \quad \text{(this will yield a very large number)} \]

**Step 3: Compare the values.**

- \(10^{11} = 100000000000\)
- \(5^{5} = 3125\) (smallest)
- \(12^{9}\) is significantly larger than \(10^{11}\) and \(5^{5}\).
- \(11^{10}\) is also very large, but we need to compare it directly with \(12^{9}\).

**Step 4: Final comparison.**

To compare \(12^{9}\) and \(11^{10}\), we can estimate:
- \(12^{9} \approx 12^{9} = 5.15978 \times 10^{8}\)
- \(11^{10} \approx 11^{10} = 2.839 \times 10^{9}\)

From the calculations, we can see that:
- \(12^{9}\) is smaller than \(11^{10}\).

**Conclusion:**

After evaluating all options, we find that:
- \(11^{10}\) has the highest value.

Thus, the answer is **(d) 11 power 10**.

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