Powers: Identify the greater value

Let's solve the problem by evaluating each expression:

• Calculate $2^4$:
  $2^4 = 2 \times 2 \times 2 \times 2 = 16$.
• Calculate $3^7$:
  $3^7 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 2187$.
• Calculate $10^2$:
  $10^2 = 10 \times 10 = 100$.
• Calculate $5^5$:
  $5^5 = 5 \times 5 \times 5 \times 5 \times 5 = 3125$.

Now, let's compare the computed values: $16$ (for $2^4$), $2187$ (for $3^7$), $100$ (for $10^2$), and $3125$ (for $5^5$).

Clearly, the value $3125$ is the largest among the values we've calculated.

Therefore, the expression with the greatest value is $5^5$.

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 $7^4$.

To solve this problem, we'll compare the two expressions $6^1$ and $1^6$ by computing each power:

• Step 1: Calculate $6^1$.
• Since the exponent is 1, $6^1 = 6$.
• Step 2: Calculate $1^6$.
• Any number raised to a power of 6 is multiplied by itself six times. Here, $1^6 = 1 \times 1 \times 1 \times 1 \times 1 \times 1 = 1$.
• Step 3: Compare the results.
• $6^1 = 6$ and $1^6 = 1$.

Since $6 > 1$, we conclude that $6^1$ is larger than $1^6$.

Therefore, the answer is $>$.

To solve this problem, we will follow these steps:

• Step 1: Calculate $5^2$.
• Step 2: Calculate $2^5$.
• Step 3: Compare the results to determine which is larger.

Let's work through each step:
Step 1: Calculate $5^2$.
$5^2 = 5 \times 5 = 25$.

Step 2: Calculate $2^5$.
$2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$.

Step 3: Compare the results.
We have $5^2 = 25$ and $2^5 = 32$. Clearly, $25 < 32$.

Therefore, $5^2 \text{ }_{——} 2^5$ is < .

To solve this problem, we'll follow these steps:

• Step 1: Evaluate $0^{100}$.
• Step 2: Evaluate $100^0$.
• Step 3: Compare the values obtained in Step 1 and Step 2.

Now, let's work through each step:

Step 1: Evaluate $0^{100}$.
Any non-negative integer power of 0 evaluates to 0. Therefore, $0^{100} = 0$.

Step 2: Evaluate $100^0$.
By the zero exponent rule for non-zero bases, $100^0 = 1$.

Step 3: Compare the values obtained: $0$ and $1$.
Clearly, $0 < 1$.

Therefore, $0^{100}$ is less than $100^0$.

The correct choice is: <

To solve this problem, we shall compute the value of each expression:

• First, calculate $12^4$:
  $12^4 = 12 \times 12 \times 12 \times 12 = 20736$.
• Next, calculate $9^4$:
  $9^4 = 9 \times 9 \times 9 \times 9 = 6561$.
• Then, calculate $5^6$:
  $5^6 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 15625$.
• Finally, calculate $3^5$:
  $3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243$.

After computing the values, we compare them:

• $12^4 = 20736$
• $9^4 = 6561$
• $5^6 = 15625$
• $3^5 = 243$

Among these, the highest value is $20736$, which corresponds to $12^4$.

Therefore, the expression with the highest value is $12^4$.

To solve this problem, we'll follow these steps:

• Step 1: Recognize that $\frac{1}{2}$ is equivalent to $0.5$.

• Step 2: Evaluate both expressions using this equivalence.

• Step 3: Conclude based on the equality of the expressions.

Now, let's work through each step:
Step 1: It's important to understand the equivalence between $\frac{1}{2}$ and $0.5$. As a fraction, $\frac{1}{2}$ is equal to the decimal $0.5$.
Step 2: We apply the power to each base: $(\frac{1}{2})^5$ and $(0.5)^5$. Due to their equivalence, $(\frac{1}{2})^5$ is necessarily equal to $(0.5)^5$.
Step 3: Since both expressions compute to the same value because their bases are identical ($\frac{1}{2} = 0.5$), the two expressions are equal.

Therefore, the solution to the problem is$=$.

To solve this problem, we will calculate $2^4$ and $4^2$ and then compare the results:

• Step 1: Calculate $2^4$
  $2^4$ means multiplying 2 by itself 4 times: $2 \times 2 \times 2 \times 2 = 16$.
• Step 2: Calculate $4^2$
  $4^2$ means multiplying 4 by itself 2 times: $4 \times 4 = 16$.
• Step 3: Compare the results
  Since both calculations result in 16, we have $2^4 = 4^2$.

Therefore, the correct comparison is that $2^4$ is equal to $4^2$. The answer to the problem is $=$.

Let's solve the problem by calculating each value:

Step 1: Calculate $7^2$.
$7^2 = 7 \times 7 = 49$.

Step 2: Calculate $7^3$.
$7^3 = 7 \times 7 \times 7 = 343$.

Step 3: Compare $49$ and $343$.
We can clearly see that $49$ is less than $343$.

Therefore, we have $7^2 < 7^3$.

The correct comparison sign is $<$.

Thus, choice 1 is correct: $<$.

To determine which of the two expressions is larger, we need to evaluate them using a comparability method:

• Step 1: Observe that these are powers of a common base. We first convert $(0.3)^8$ to a fraction form. The number $0.3$ is equivalent to $\frac{3}{10}$.
• Step 2: Thus, $(0.3)^8$ becomes $\left(\frac{3}{10}\right)^8$.
• Step 3: We then compare $\left(\frac{3}{10}\right)^7$ and $\left(\frac{3}{10}\right)^8$. Since both have the same base, $\frac{3}{10}$, we compare the exponents directly: $7$ and $8$.
• Step 4: Clearly, any number to the power of 7 will be larger than the same number to the power of 8, since raising a positive fraction to a higher power results in a smaller number.

This analysis shows that $\left(\frac{3}{10}\right)^7$ is larger than $(0.3)^8$. Therefore, using the symbol for comparison:

The correct comparison is (\frac{3}{10})^7 > (0.3)^8 .

Accordingly, the solution to this problem as stated is incorrect in the given answer. The correct comparison should indeed be:

$(\frac{3}{10})^7 > (0.3)^8$

The problem involves comparing $16^8$ and $16^{(2+6)}$.

First, simplify the exponent in the second expression:

• $16^{(2+6)} = 16^8$

This simplifies directly to $16^8$, which is identical to the first expression, $16^8$.

Since both expressions are equal after simplification, we conclude that:

The two expressions are equal, therefore $=$.