Powers: Complete the missing number

To solve this problem, we have to determine the missing number in the expression $☐^6 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$.

Let's follow these steps:

• Step 1: Recognize that the right side of the equation, $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$, is equivalent to $2^6$, as it involves multiplying six 2s together.
• Step 2: The left side of the equation $☐^6$ means some number raised to the power of 6.
• Step 3: Since the two sides must be equal, we recognize that the base on both sides must be the same if the exponents are equal. Hence, the missing base number, represented by $☐$, is $2$.

When we match the two expressions based on exponents, we find that the correct base completing the equation $☐^6 = 2^6$ is $2$.

Therefore, the missing number is $\mathbf{2}$.

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

• Step 1: Evaluate the power expression $5 \cdot 5 \cdot 5$
• Step 2: Determine the number whose cube matches this evaluated expression
• Step 3: Conclude with the missing number that satisfies $☐^3 = 5 \cdot 5 \cdot 5$

Now, let's work through these steps:

Step 1: Calculate the power expression.

$5 \cdot 5 \cdot 5 = 125$

Step 2: Determine the cube root of 125 to find the missing number.

Since $5^3 = 125$, it follows that the missing number must be 5.

Step 3: Verify that $5^3 = 125$.

Therefore, the solution to the given problem is $5$.

To solve this problem, we need to understand how powers with a base of zero work. Typically, for any positive integer $n$, raising zero to that power results in zero, as follows:

• $0^n = 0$ for $n > 0$.

Therefore, to satisfy the equation $0^\square = 0$, the exponent $\square$ should be any positive integer. Hence, the missing number that makes the equation true is simply any positive integer.

Therefore, the correct answer is a (any number), which corresponds to any positive integer number.

The missing number is 3.

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

• Step 1: Identify how many times 4 is multiplied by itself in the expression $4 \cdot 4 \cdot 4 \cdot 4$.
• Step 2: Determine the exponent that matches this count.

Now, let's work through each step:
Step 1: In the expression $4 \cdot 4 \cdot 4 \cdot 4$, the base number 4 is multiplied by itself 4 times.
Step 2: According to the rule of exponents, $a^n$ is the notation for a number $a$ multiplied by itself $n$ times.

Therefore, the correct exponent for $4$ in the expression is 4, so the missing number is $\mathbf{4}$. The complete expression is $4^4$.

Therefore, the solution to the problem is $4$.

To solve this problem, we'll begin by simplifying the expression on the right side of the equation:

$☐^7 = \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5}$

Using the laws of exponents, multiplying the fraction $\frac{1}{5}$ by itself seven times can be expressed as:

$\left(\frac{1}{5}\right)^7$

Now, the equation becomes:

$☐^7 = \left(\frac{1}{5}\right)^7$

Since the exponents on both sides of the equation are the same, the bases must be equal as well. Therefore, $☐ = \frac{1}{5}$.

Thus, the missing number is:

$\frac{1}{5}$

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

• Step 1: Identify potential values for the missing number $\square$.
• Step 2: Verify if it fits the equation $☐^7 = 1$.
• Step 3: Consider alternative numbers from intuitive mathematics.

Now, let's work through each step:
Step 1: We hypothesize that the number could be 1, given that raising 1 to any power yields 1.
Step 2: Verify $(1)^7 = 1$, which is true since any real number 1 raised to any integer power remains 1.
Step 3: As a check for understanding: with odd powers greater than 1, attempts with $-1$ as choices lead to $(-1)^7 = -1$, which doesn’t meet the requirement, reinforcing $☐ = 1$.

Therefore, the solution to the problem is 1, choice number 4.

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

• Step 1: Count the number of times $\frac{1}{5}$ is used as a factor in the product given.
• Step 2: Use this count as the exponent in the expression $\frac{1}{5}^☐$.

Now, let's work through each step:
Step 1: The expression $\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5}$ shows that the base $\frac{1}{5}$ is used 5 times.
Step 2: Therefore, the exponent that makes the expression $\left( \frac{1}{5} \right)^☐$ equal to the product is 5.

Therefore, the missing number in the expression $\frac{1}{5}^☐$ is $5$.

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

• Step 1: Identify the structure of the exponentiation in the problem.
• Step 2: Count the number of terms multiplied on the right-hand side.
• Step 3: Equate the number of terms to the exponent, thereby finding the missing number.

Now, let's work through each step:

Step 1: The problem provides the expression: $\frac{1}{a}^☐$ on the left-hand side, and $\frac{1}{a} \cdot \frac{1}{a} \cdot \frac{1}{a} \cdot \frac{1}{a} \cdot \frac{1}{a} \cdot \frac{1}{a}$ on the right-hand side.

Step 2: Count the number of $\frac{1}{a}$ terms on the right. There are 6 terms.

Step 3: The property of exponents allows us to say $\left(\frac{1}{a}\right)^☐$ should equal to $\frac{1}{a}$ multiplied by itself 6 times. Thus, the exponent on the left, indicated by ☐, must match the count of the terms:

Therefore, the missing number for ☐ is $6$.

To solve this problem, we'll adopt the following method:

• Step 1: Understand the given expression in multiplication form: $x \cdot x \cdot x \cdot x \cdot x \cdot x$.
• Step 2: Count the number of times $x$ appears.
• Step 3: Determine that the expression is equivalent to $x^n$, where $n$ is the count of $x$ factors.

Now, let's work through these steps:

Step 1: We are given the product $x \cdot x \cdot x \cdot x \cdot x \cdot x$, which is clearly a multiplication of $x$ six times.

Step 2: Counting these, we see that $x$ appears 6 times.

Step 3: Therefore, the product can be expressed as $x^6$, meaning that the expression $x^☐ = x \cdot x \cdot x \cdot x \cdot x \cdot x$ implies that $☑ = 6$.

Thus, the missing number is 6, corresponding to choice 2.