Compare Powers: Finding the Largest Value Among 7⁴, 8³, 10², and 11^(1/2)

Which of the expressions below has the highest value?

$7^4,8^3,10^2,11^{\frac{1}{2}}$

The goal is to compare the values of the expressions $7^4$, $8^3$, $10^2$, and $11^{\frac{1}{2}}$ to identify the one with the highest value.

Let's calculate each expression:

• Calculate $7^4$:
  - $7^4 = 7 \times 7 \times 7 \times 7$
  - $7 \times 7 = 49$
  - $49 \times 7 = 343$
  - $343 \times 7 = 2401$
  Therefore, $7^4 = 2401$.
• Calculate $8^3$:
  - $8^3 = 8 \times 8 \times 8$
  - $8 \times 8 = 64$
  - $64 \times 8 = 512$
  Therefore, $8^3 = 512$.
• Calculate $10^2$:
  - $10^2 = 10 \times 10 = 100$.
  Therefore, $10^2 = 100$.
• Calculate $11^{\frac{1}{2}}$ (the square root of 11):
  - $11^{\frac{1}{2}} \approx \sqrt{11} \approx 3.3166$.
  Therefore, $11^{\frac{1}{2}} \approx 3.3166$.

Now, let's compare these values:

• $7^4 = 2401$
• $8^3 = 512$
• $10^2 = 100$
• $11^{\frac{1}{2}} \approx 3.3166$

Clearly, $2401$ is the highest value among these, which is obtained from $7^4$.

Therefore, the expression with the highest value is $7^4$.