Powers and Roots - Basic - Examples, Exercises and Solutions

To solve this problem, we'll focus on expressing the power $2^7$ as a series of multiplications.

• Step 1: Identify the given power expression $2^7$.
• Step 2: Convert $2^7$ into a product of repeated multiplication. This involves writing 2 multiplied by itself for a total of 7 times.
• Step 3: The expanded form of $2^7$ is $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$.

By comparing this expanded form with the provided choices, we see that the correct expression is:

$2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2$

Therefore, the solution to the problem is the expression that matches this expanded multiplication form, which is the choice $\text{1: } 2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2$.

To solve this problem, we will apply the rule of exponents:

• Any number raised to the power of 1 remains unchanged. Therefore, by the identity property of exponents, $10,000^1 = 10,000$.

Given the choices:

• $10,000 \cdot 10,000$: This is $10,000^2$.
• $10,000 \cdot 1$: Simplifying this expression yields 10,000, which is equivalent to $10,000^1$.
• $10,000 + 10,000$: This results in 20,000, not equivalent to $10,000^1$.
• $10,000 - 10,000$: This results in 0, not equivalent to $10,000^1$.

Therefore, the correct choice is $10,000 \cdot 1$, which simplifies to 10,000, making it equivalent to $10,000^1$.

Thus, the expression $10,000^1$ is equivalent to:

$10,000 \cdot 1$

To solve this problem, we'll determine the square root of the number 4.

• Step 1: Recognize that the square root of a number is asking for a value that, when multiplied by itself, yields the original number. Here, we seek a number $y$ such that $y^2 = 4$.
• Step 2: Identify that $4$ is a perfect square. The numbers $2$ and $-2$ both satisfy the equation $2^2 = 4$ and $(-2)^2 = 4$.
• Step 3: We usually consider the principal square root, which is the non-negative version. Thus, $\sqrt{4} = 2$.

Therefore, the solution to the problem is 2, which corresponds to the correct choice from the given options.

To solve this problem, we want to find the square root of 9.

Step 1: Recognize that a square root is a number which, when multiplied by itself, equals the original number. Thus, we are seeking a number $x$ such that $x^2 = 9$.

Step 2: Note that 9 is a common perfect square: $9 = 3 \times 3$. Therefore, the square root of 9 is the number that, when multiplied by itself, gives 9. This number is 3.

Step 3: Since we are interested in the principal square root, we consider only the non-negative value. Hence, the principal square root of 9 is 3.

Therefore, the solution to the problem is $3$.

To determine the square root of 16, follow these steps:

• Identify that we are looking for the square root of 16, which is a number that, when multiplied by itself, equals 16.
• Recall the basic property of perfect squares: $4 \times 4 = 16$.
• Thus, the square root of 16 is 4.

Hence, the solution to the problem is the principal square root, which is $4$.

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

• Step 1: Understand the definition of a square root.
• Step 2: Identify which integer, when squared, gives 36.
• Step 3: Verify this integer meets the required condition.
• Step 4: Choose the correct answer from the given choices.

Now, let's work through each step:
Step 1: A square root of a number is a value that, when multiplied by itself, gives the original number. Here, we want $y$ such that $y^2 = 36$.
Step 2: We test integer values to find which one squared equals 36. Testing $y = 1, 2, 3, 4, 5,$ and $6$ gives:
- $1^2 = 1$
- $2^2 = 4$
- $3^2 = 9$
- $4^2 = 16$
- $5^2 = 25$
- $6^2 = 36$

Step 3: The integer $6$ satisfies $6^2 = 36$. Therefore, $\sqrt{36} = 6$.

Step 4: The correct choice from the given answer choices is 6 (Choice 4).

Hence, the square root of 36 is $\mathbf{6}$.

To solve this problem, we follow these steps:

• Step 1: Understand that finding the square root of a number means determining what number, when multiplied by itself, equals the original number.
• Step 2: Identify the numbers that could potentially be the square root of $49$. These are $\pm7$, but by convention, the square root function typically refers to the non-negative root.
• Step 3: Calculate $7 \times 7 = 49$. This confirms that $\sqrt{49} = 7$.
• Step 4: Verify using the problem's multiple-choice answers to ensure $7$ is among them, confirming choice number .

Therefore, the solution to the problem $\sqrt{49}$ is $7$.

To solve this problem, we'll determine the square root of 64, following these steps:

• Step 1: Identify the number whose square root we need to find. The number given is 64.
• Step 2: Determine which number, when multiplied by itself, equals 64.
• Step 3: Recall that $8 \times 8 = 64$.

Now, let's work through each step:

Step 1: We are tasked with finding $\sqrt{64}$. The problem involves identifying the number which, when squared, results in 64.
Step 2: To find this number, we'll check our knowledge of squares. We know that 8 is a significant integer whose square results in 64.
Step 3: Compute: $8 \times 8 = 64$. Hence, $8$ meets the requirement.

We find that the solution to the problem is $\sqrt{64} = 8$.

To solve this problem, follow these steps:

• Step 1: Understand that the square root of a number $n$ is a value that, when multiplied by itself, equals $n$.
• Step 2: Identify the number whose square is 81. Since $9 \times 9 = 81$, the square root of 81 is 9.

Therefore, the square root of 81 is $9$.

The task is to find the square root of the number 100. The square root operation seeks a number which, when squared, equals the original number. For any positive integer, if $x^2 = 100$, then $x$ should be our answer.

Step 1: Recognize that 100 is a perfect square. This means there exists an integer $x$ such that $x \times x = 100$. Generally, we recall basic squares such as:

• $1^2 = 1$
• $2^2 = 4$
• $3^2 = 9$
• and so forth, up to $10^2$

Step 2: Checking integers, we find that:

$10^2 = 10 \times 10 = 100$

Step 3: Confirm the result: Since $10 \times 10 = 100$, then $\sqrt{100} = 10$.

Step 4: Compare with answer choices. Given that one of the choices is 10, and $\sqrt{100} = 10$, choice 1 is correct.

Therefore, the square root of 100 is 10.

To solve this problem, we need to determine the square root of 25.

• Step 1: The square root operation asks us to find a number that, when multiplied by itself, equals the given number, 25.
• Step 2: Consider what number times itself equals 25. We note that $5 \times 5 = 25$.
• Step 3: Thus, the square root of 25 is 5.

Therefore, the solution to the problem is $\sqrt{25} = 5$.

The correct answer is choice 2: 5.

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

• Step 1: Recognize that $6^2$ means $6 \times 6$.
• Step 2: Perform the multiplication of 6 by itself.

Now, let's work through each step:
Step 1: The expression $6^2$ indicates we need to multiply 6 by itself.
Step 2: Calculating $6 \times 6$ gives us 36.

Therefore, the value of $6^2$ is 36.

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

• Step 1: Set up the multiplication as $11 \times 11$.
• Step 2: Compute the product using basic arithmetic.
• Step 3: Compare the result with the provided multiple-choice answers to identify the correct one.

Now, let's work through each step:
Step 1: We begin with the calculation $11 \times 11$.
Step 2: Perform the multiplication:

1. Multiply the units digits: $1 \times 1 = 1$.
2. Next, for the tens digits: $11 \times 10 = 110$.
3. Add the results: $110 + 1 = 111$. This doesn't seem right, so let's break it down further.

Let's examine a more structured multiplication method:

Multiply $11$ by $1$ (last digit of the second 11), we get 11.
Multiply $11$ by $10$ (tens place of the second 11), we get 110.

If we align correctly and add the partial products:

     11
+   110
------------
   121

Step 3: The correct multiplication yields the final result as $121$. Upon reviewing the provided choices, the correct answer is choice 4: $121$.

Therefore, the solution to the problem is $121$.

Let's solve the problem step by step:

• Step 1: Understand what the square root means.

The square root of a number $n$ is a value that, when multiplied by itself, equals $n$. This is written as $x = \sqrt{n}$.

• Step 2: Apply this definition to the number $36$.

We are looking for a number $x$ such that $x^2 = 36$. This translates to finding $x = \sqrt{36}$.

• Step 3: Determine the correct number.

We know that $6 \times 6 = 36$. Therefore, the principal square root of $36$ is $6$.

Thus, the solution to the problem is $\sqrt{36} = 6$.

Among the given choices, the correct one is: Choice 1: $6$.

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

• Step 1: Recognize what finding a square root means
• Step 2: List known perfect squares to identify which one results in 64
• Step 3: Verify the square root by calculation

Now, let's work through each step:
Step 1: To find the square root of 64, we seek a number that, when multiplied by itself, equals 64.
Step 2: Consider the sequence of perfect squares: $1^2 = 1$, $2^2 = 4$, $3^2 = 9$, $4^2 = 16$, $5^2 = 25$, $6^2 = 36$, $7^2 = 49$, $8^2 = 64$.
Step 3: We see that $8^2 = 64$. Therefore, the square root of 64 is 8.

Therefore, the solution to this problem is $8$.