Solve the following exercise:
Solve the following exercise:
\( \sqrt{\sqrt{2}}= \)
Solve the following exercise:
\( \sqrt[5]{\sqrt[3]{5}}= \)
Solve the following exercise:
\( \sqrt[10]{\sqrt[10]{1}}= \)
Solve the following exercise:
\( \sqrt[6]{\sqrt{2}}= \)
Solve the following exercise:
\( \sqrt[]{\sqrt{8}}= \)
Solve the following exercise:
To solve , we will use the property of roots.
Therefore, .
This corresponds to choice 2: .
The solution to the problem is .
Solve the following exercise:
To solve the problem of finding , we'll use the formula for a root of a root, which combines the exponents:
Therefore, the simplified expression of is .
Solve the following exercise:
To solve this problem, we'll observe the following process:
Thus, the evaluation of the original expression equals 1.
Comparing this result to the provided choices:
Therefore, choice 4 is correct: All answers are equivalent to the solution, being 1.
Thus, the correct selection is: All answers are correct.
All answers are correct.
Solve the following exercise:
Express the definition of root as a power:
Remember that in a square root (also called "root to the power of 2") we don't write the root's power as shown below:
Meaning:
Now convert the roots in the problem using the root definition provided above. :
In the first stage we applied the root definition as a power mentioned earlier to the inner expression (meaning inside the larger-outer root) and then we used parentheses and applied the same definition to the outer root.
Let's recall the power law for power of a power:
Apply this law to the expression that we obtained in the last stage:
In the first stage we applied the power law mentioned above and then proceeded first to simplify the resulting expression and then to perform the multiplication of fractions in the power exponent.
Let's summarize the various steps of the solution thus far:
In the next stage we'll apply once again the root definition as a power, (that was mentioned at the beginning of the solution) however this time in the opposite direction:
Let's apply this law in order to present the expression we obtained in the last stage in root form:
We obtain the following result: :
Therefore the correct answer is answer A.
Solve the following exercise:
In order to solve the given problem, we'll follow these steps:
Step 1: Convert the inner square root to an exponent: .
Step 2: Apply the root of a root property: .
Step 3: Simplify the expression using exponent rules: .
The nested root expression simplifies to .
Therefore, the simplified expression of is .
After comparing this result with the multiple choice answers, choice 2 is correct.
Solve the following exercise:
\( \sqrt[6]{\sqrt{2}}= \)
Solve the following exercise:
\( \sqrt{\sqrt{12}}= \)
Solve the following exercise:
\( \sqrt{\sqrt{4}}= \)
Solve the following exercise:
\( \sqrt[7]{\sqrt{2}}= \)
Solve the following exercise:
\( \sqrt[3]{\sqrt{729}}= \)
Solve the following exercise:
In order to solve this problem, we must simplify the following expression using the rule for roots of roots. This rule states that a root of a root can be written as a single root by multiplying the indices of the radicals.
Step 1: Identify the given expression .
Step 2: Recognize that the inner root, , can be expressed as .
Step 3: Visualize as .
Step 4: Apply the rule .
Step 5: Multiply the indices: .
Step 6: Replace the compound root with the single root: .
Thus, the expression simplifies to .
Therefore, the solution to the problem is .
Solve the following exercise:
In order to solve the following expression , it needs to be simplified using the properties of exponents and roots. Specifically, we apply the rule that states that the square root of a square root can be expressed as a fourth root.
Let's break down this solution step by step:
First, represent the inner as a power: .
Next, take the square root of this result, which involves raising to the power of again:
.
According to the rules of exponents, raising an exponent to another power results in multiplying the exponents.
This gives us , which we can write as the fourth root of 12: .
In conclusion the simplification of is .
Solve the following exercise:
To solve the expression , we'll proceed with the following steps:
Therefore, the answer to the problem is .
Solve the following exercise:
Let's solve the given problem by following these steps:
Thus, the simplified expression is .
Therefore, the solution to the problem is .
Solve the following exercise:
To solve this problem, we need to evaluate .
Step 1: Convert the expression into exponent form. We know . Thus, .
Step 2: Apply the formula for exponents . Thus, .
Step 3: Find the base of 729 as a power of an integer. Observing, 729 = because .
Step 4: Substitute the power of the base: .
Therefore, the solution to the problem is .
3
Solve the following exercise:
\( \sqrt[3]{\sqrt[3]{512}}= \)
Solve the following exercise:
\( \sqrt[5]{\sqrt[]{1024}}= \)
Solve the following exercise:
\( \sqrt{\sqrt{625}}= \)
Solve the following exercise:
\( \sqrt[3]{\sqrt{64}}= \)
Solve the following exercise:
\( \sqrt[3]{\sqrt{144}}= \)
Solve the following exercise:
To solve this problem, we'll proceed with the following steps:
Therefore, the solution to the expression is .
2
Solve the following exercise:
To solve this problem, we'll follow these steps:
Now, let's work through each step:
Step 1: Calculate the inner square root.
We have . We know that , so .
Applying the property of roots, .
Step 2: Now, apply the fifth root to the result from step 1.
We need to find .
Step 3: Simplify using the properties of exponents.
From , we have .
Therefore, the solution to the problem is .
2
Solve the following exercise:
To solve this problem, we'll follow these steps:
Now, let's work through each step:
Step 1: Evaluate .
The square root of 625 is 25, since . Thus, .
Step 2: Evaluate .
The square root of 25 is 5, since . Thus, .
Therefore, the solution to the problem is 5.
5
Solve the following exercise:
To solve this problem, we'll express the nested roots using exponents and then simplify:
We start with the inner expression:
is equivalent to .
Next, apply the cube root:
is equivalent to .
Using properties of exponents, we simplify the expression:
.
Now, evaluate :
Since , we have:
.
Therefore, the solution to the exercise is .
2
Solve the following exercise:
To solve this problem, let's follow these steps:
Let's apply these steps:
Step 1: The square root of 144 can be expressed as .
Step 2: We need the cube root of this expression, so we have .
Step 3: Using the property of exponents , we multiply the exponents: .
Step 4: Re-express this as a root: Since is equivalent to the sixth root, we have .
Therefore, the solution to the problem is , which corresponds to choice 3.
Solve the following exercise:
\( \sqrt[7]{\sqrt[4]{14}}= \)
Solve the following exercise:
\( \sqrt[4]{\sqrt[3]{16}}= \)
Solve the following exercise:
\( \sqrt[4]{\sqrt{6}}= \)
Solve the following exercise:
\( \sqrt[10]{\sqrt[5]{100}}= \)
Complete the following exercise:
\( \sqrt[3]{\sqrt{36}}= \)
Solve the following exercise:
To solve this problem, we'll follow these steps:
Now, let's work through each step:
Step 1: The expression given is . The inner root is , and the outer root is .
Step 2: Use the formula for the root of a root: .
Step 3: Plug in our values: and . Thus, we have:
Therefore, the simplified form of the given expression is .
Solve the following exercise:
To solve this problem, we'll convert the roots into exponents and simplify:
Now, let's work through each step:
Step 1: The cube root of 16 can be written as . Thus, our expression becomes .
Step 2: Apply the fourth root, which is an exponent of . This gives us .
Step 3: From the original question, the expression simplifies to , which is equivalent to . Therefore, the choices that are correct are the ones that reflect this equivalence.
Therefore, the solution to the problem involves recognizing that both and represent the same value, and thus, answers a and b are correct.
Answers a and b are correct
Solve the following exercise:
To solve , we'll simplify it using the property of roots:
According to the property , we are to multiply the indices of the roots.
Here, the indices are 4 (for the fourth root) and 2 (for the square root), hence and .
Therefore, we calculate:
This indicates that the expression can be rewritten as a single root of index 8, giving us:
The answer is therefore , which matches answer choice 4.
Solve the following exercise:
To solve this problem, we'll follow these steps:
Let's apply these steps to find the solution:
First, we apply the root of a root property:
This simplifies to:
Therefore, the solution to the problem is .
Complete the following exercise:
To solve this problem, let's analyze and simplify the given expression .
Step 1: Identify the root operations. We have a square root, , and a cube root, .
Step 2: Use the formula for roots for a root of a root: .
Step 3: Apply this formula to the problem. In this case, the first operation is a square root, which can be written as , and the second operation is a cube root. Therefore, .
Step 4: Simplify using the power of a power rule, which allows us to multiply exponents: .
Thus, the expression simplifies to .