Powers: Simple exercise

To solve this problem, we need to calculate the value of $8^3$.

Let's work through the calculation step-by-step:

• First, calculate $8 \times 8$. This gives: $8 \times 8 = 64$ .
• Next, we multiply the result by 8 again: $64 \times 8 = 512$ .

Therefore, the value of $8^3$ is 512.

To solve the problem of finding $6^5$ and expressing it as an integer, we need to carry out the multiplication of $6$ by itself five times. Here is the step-by-step calculation:

• Step 1: Multiply $6 \times 6$ which equals $36$.
• Step 2: Multiply $36 \times 6$ which results in $216$.
• Step 3: Multiply $216 \times 6$ obtaining $1296$.
• Step 4: Multiply $1296 \times 6$ to get $7776$.

By following these calculations, we find that $6^5 = 7776$.

Thus, the integer representation of $6^5$ is $7776$.

To solve the problem of determining $4^4$, we follow a straightforward calculation of powers:

Step 1: Understand the expression $4^4$ which means $4$ multiplied by itself 4 times.

Step 2: Calculate:

• First multiplication: $4 \times 4 = 16$.
• Second multiplication: $16 \times 4 = 64$.
• Third multiplication: $64 \times 4 = 256$.

Thus, $4^4$ results in an integer value of 256.

Therefore, the solution to the problem is $256$.

To solve this problem, we need to evaluate the expression $7^2$. This expression means $7$ multiplied by itself, due to the exponent $2$.

• Step 1: Identify the base and exponent — here the base is $7$ and the exponent is $2$.
• Step 2: Apply the definition of exponents: calculate $7 \times 7$.

Now, let's perform the calculation:

$7 \times 7 = 49$.

Therefore, the integer value of $7^2$ is $\textbf{49}$.

To solve the problem of finding $5^3$, we will proceed as follows:

• Step 1: Identify that $5^3$ means $5$ raised to the power of 3, which is equivalent to multiplying 5 by itself 3 times.
• Step 2: Compute the expression: $5^3 = 5 \times 5 \times 5$.
• Step 3: Perform the multiplication in two stages for clarity:
• First, calculate $5 \times 5 = 25$.
• Next, take the result from the first calculation and multiply by 5 again: $25 \times 5 = 125$.

Therefore, the value of $5^3$ is 125.

This matches with one of the provided choices, which is 125.

To solve this problem, we will compute $16^3$ by performing the following steps:

• First, calculate the square of 16: $16 \times 16 = 256$.
• Next, use the result to find $16^3$: $256 \times 16 = 4096$.

Therefore, the value of $16^3$ is $4096$.

To solve this problem, we'll perform repeated multiplication to find $12^5$, as follows:

• Step 1: Compute $12^2 = 12 \times 12 = 144$.

• Step 2: Compute $12^3 = 144 \times 12 = 1728$.

• Step 3: Compute $12^4 = 1728 \times 12 = 20736$.

• Step 4: Finally, compute $12^5 = 20736 \times 12 = 248832$.

Through these calculations, we determined that the value of $12^5$ is $248832$.

Therefore, the solution to the problem is $248832$ and this corresponds to choice 3.

To solve this problem, we will perform step-by-step calculations to find $13^4$:

• Step 1: Calculate $13^2$. We have $13 \times 13 = 169$.
• Step 2: Calculate $13^3$ using the result from Step 1. We multiply $169 \times 13$. Performing the multiplication:
  $169 \times 13 = 169 \times (10 + 3) = 169 \times 10 + 169 \times 3 = 1690 + 507 = 2197$.
• Step 3: Calculate $13^4$ using the result from Step 2. Multiply $2197 \times 13$. Performing the multiplication:
  $2197 \times 13 = 2197 \times (10 + 3) = 2197 \times 10 + 2197 \times 3 = 21970 + 6591 = 28561$.

Therefore, the value of $13^4$ is 28561, which corresponds to choice (2).

To solve this problem, we'll multiply as follows:

• Step 1: Compute $11 \times 11 = 121$.

• Step 2: Take the result from Step 1 and multiply it by 11 again: $121 \times 11$.

Let's perform the multiplication for Step 2:

To calculate $121 \times 11$, we can use a standard multiplication method:

• 121
  ×11
  ----

• From the rightmost digit: $1 \times 1 = 1$.

• Next: $1 \times 2 = 2$. Add the carried over part (none here, so just 2).

• Finally: $1 \times 1 = 1$.

Thus, the basic multiplication gives us 121. Now multiply by 10 (shift left):

• Shifted 121 becomes 1210 (since it represents $121 \times 10$).

• Add these values:
  121
  +1210
  -----
  1331

Therefore, the solution to $11^3$ is 1331.

To solve the problem of finding $10^3$, let us apply the rules of exponents:

• Step 1: Recognize that $10^3$ means the base 10 is multiplied by itself three times.
• Step 2: Write out this multiplication: $10 \times 10 \times 10$.
• Step 3: Calculate the result step by step.
  First, calculate $10 \times 10 = 100$.
  Next, multiply the result by 10: $100 \times 10 = 1000$.

Therefore, the expression $10^3$ evaluates to 1000.

Thus, the solution to this problem is $1000$, which corresponds to choice 4.