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 $\text{1: } 2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2$.

Let's begin by calculating the numerical value of each of the roots in the given options:

$\sqrt{25}=5\\ \sqrt{16}=4\\ \sqrt{9}=3\\$We can determine that:

5>4>3>1 Therefore, the correct answer is option A

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 $121$.

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.

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 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 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 $8$.

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:

$10,000 \cdot 1$

We use the formula: $a\times a=a^2$

In the formula, we see that the power shows the number of terms that are multiplied, that is, two times

Since in the exercise we multiply 6 three times, it means that we have 3 terms.

Therefore, the power, which is n in this case, will be 3.

In order to simplify the given expression, we will use two laws of exponents:

a. The definition of root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$b. The law of exponents for power of a power:

$(a^m)^n=a^{m\cdot n}$

Let's start with converting the square root to an exponent using the law mentioned in a':

$\sqrt{x^2}= \\ \downarrow\\ (x^2)^{\frac{1}{2}}=$We'll continue using the law of exponents mentioned in b' and perform the exponent operation on the term in parentheses:

$(x^2)^{\frac{1}{2}}= \\ x^{2\cdot\frac{1}{2}}\\ x^1=\\ \boxed{x}$Therefore, the correct answer is answer a'.

Remember that according to the order of operations, exponents precede multiplication and division, which precede addition and subtraction (and parentheses always precede everything).

So first calculate the values of the terms in the power and then subtract between the results:

$3^2+3^3 =9+27=36$Therefore, the correct answer is option B.

To solve this problem, we'll evaluate $5^3$, which means we need to calculate the product of multiplying the number $5$ by itself three times.

• Step 1: Compute $5 \times 5$.
• Step 2: Compute the result of $5 \times 5$ and multiply it by $5$ again.

Let's work through each step:

Step 1: Calculate $5 \times 5 = 25$.

Step 2: Take the result $25$ and multiply it by $5$:
$25 \times 5 = 125$.

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

To solve the expression $5+\sqrt{36}-1=$, we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Here are the steps:

First, calculate the square root:

$\sqrt{36} = 6$

Substitute the square root back into the expression:

$5 + 6 - 1$

Next, perform the addition and subtraction from left to right:

Add 5 and 6:

$5 + 6 = 11$

Then subtract 1:

$11 - 1 = 10$

Finally, you obtain the solution:

$10$

To solve the problem of finding $7^3$, follow these steps:

• Step 1: Understand that the power $7^3$ means 7 multiplied by itself three times.
• Step 2: First, compute $7 \times 7$. This equals 49.
• Step 3: Next, multiply the result by 7 again: $49 \times 7$.
• Step 4: Calculate $49 \times 7$, which equals 343.

Therefore, $7^3 = 343$.

With the possible choices given, the correct answer corresponds to choice $343$.

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.