When we encounter an exercise in which there is a root applied to another root, we will multiply the order of the first root by the order of the second and the order obtained (the product of both) will be raised as a root in our number (as generally power of a power) Let's put it this way:
Step 1: Recognize that 2 involves two square roots.
Step 2: Each square root can be expressed using exponents: 2=21/2.
Step 3: Therefore, 2=(21/2)1/2.
Step 4: Apply the formula for the root of a root: (xa)b=xab.
Step 5: For (21/2)1/2, this means we compute the product of the exponents: (1/2)×(1/2)=1/4.
Step 6: The expression simplifies to 21/4, which is written as 42.
Therefore, 2=42.
This corresponds to choice 2: 42.
The solution to the problem is 42.
Answer
42
Exercise #2
Solve the following exercise:
535=
Video Solution
Step-by-Step Solution
To solve the problem of finding 535, we'll use the formula for a root of a root, which combines the exponents:
Step 1: Express each root as an exponent.
We start with the innermost root: 35=51/3.
Step 2: Apply the outer root.
The square root to the fifth power is expressed as: 551/3=(51/3)1/5.
Step 3: Combine the exponents.
Using the exponent rule (am)n=am×n, we get: (51/3)1/5=5(1/3)×(1/5)=51/15.
Step 4: Convert the exponent back to root form.
This can be written as 155.
Therefore, the simplified expression of 535 is 155.
Answer
155
Exercise #3
Solve the following exercise:
10101=
Video Solution
Step-by-Step Solution
To solve this problem, we'll observe the following process:
Step 1: Recognize the expression 10101 involves nested roots.
Step 2: Apply the formula for nested roots: nmx=n⋅mx.
Step 3: Set n=10 and m=10, resulting in 10×101=1001.
Step 4: Simplify 1001. Any root of 1 is 1, as 1k=1 for any positive rational number k.
Thus, the evaluation of the original expression 10101 equals 1.
Comparing this result to the provided choices:
Choice 1 is 1.
Choice 2 is 1001, which is also 1.
Choice 3 is 1=1.
Choice 4 states all answers are correct.
Therefore, choice 4 is correct: All answers are equivalent to the solution, being 1.
Thus, the correct selection is: All answers are correct.
Answer
All answers are correct.
Exercise #4
Solve the following exercise:
62=
Video Solution
Step-by-Step Solution
Express the definition of root as a power:
na=an1
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:
n=2
Meaning:
a=2a=a21
Now convert the roots in the problem using the root definition provided above. :
62=6221=(221)61
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:
(am)n=am⋅n
Apply this law to the expression that we obtained in the last stage:
(221)61=221⋅61=22⋅61⋅1=2121
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:
62=(221)61=2121
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:
an1=na
Let's apply this law in order to present the expression we obtained in the last stage in root form:
2121=122
We obtain the following result: :
62=2121=122
Therefore the correct answer is answer A.
Answer
122
Exercise #5
Solve the following exercise:
8=
Video Solution
Step-by-Step Solution
In order to solve the given problem, we'll follow these steps:
Step 1: Convert the inner square root to an exponent: 8=81/2.
Step 2: Apply the root of a root property: 8=(8)1/2=(81/2)1/2.
Step 3: Simplify the expression using exponent rules: (81/2)1/2=8(1/2)⋅(1/2)=81/4.
The nested root expression simplifies to 81/4.
Therefore, the simplified expression of 8 is 841.
After comparing this result with the multiple choice answers, choice 2 is correct.
Answer
841
Question 1
Solve the following exercise:
\( \sqrt[6]{\sqrt{2}}= \)
Incorrect
Correct Answer:
\( \sqrt[12]{2} \)
Question 2
Solve the following exercise:
\( \sqrt{\sqrt{12}}= \)
Incorrect
Correct Answer:
\( \sqrt[4]{12} \)
Question 3
Solve the following exercise:
\( \sqrt{\sqrt{4}}= \)
Incorrect
Correct Answer:
\( \sqrt{2} \)
Question 4
Solve the following exercise:
\( \sqrt[7]{\sqrt{2}}= \)
Incorrect
Correct Answer:
\( \sqrt[14]{2} \)
Question 5
Solve the following exercise:
\( \sqrt[3]{\sqrt{729}}= \)
Incorrect
Correct Answer:
3
Exercise #6
Solve the following exercise:
62=
Video Solution
Step-by-Step Solution
In order to solve this problem, we must simplify the following expression 62 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 62.
Step 2: Recognize that the inner root, 2, can be expressed as 22.
Step 3: Visualize 62 as 622.
Step 4: Apply the rule nma=n×ma.
Step 5: Multiply the indices: 6×2=12.
Step 6: Replace the compound root with the single root: 122.
Thus, the expression 62 simplifies to 122.
Therefore, the solution to the problem is 122.
Answer
122
Exercise #7
Solve the following exercise:
12=
Video Solution
Step-by-Step Solution
In order to solve the following expression 12, 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 12 as a power: 121/2.
Next, take the square root of this result, which involves raising 121/2 to the power of 1/2 again: (121/2)1/2=12(1/2)⋅(1/2)=121/4.
According to the rules of exponents, raising an exponent to another power results in multiplying the exponents.
This gives us 121/4, which we can write as the fourth root of 12: 412.
In conclusion the simplification of 12 is 412.
Answer
412
Exercise #8
Solve the following exercise:
4=
Video Solution
Step-by-Step Solution
To solve the expression 4, we'll proceed with the following steps:
Step 1: Evaluate the inner square root. The expression 4 simplifies to 2, because 2 squared is 4.
Step 2: Now evaluate the square root of 2. Since the result from step 1 is 2, we need to find 2. This is the prime representation of the result because 2 cannot be further simplified.
Therefore, the answer to the problem 4 is 2.
Answer
2
Exercise #9
Solve the following exercise:
72=
Video Solution
Step-by-Step Solution
Let's solve the given problem by following these steps:
Step 1: Recognize the expression 72. It involves two roots.
Step 2: Rewrite each part using rational exponents. We have 2=21/2.
Step 3: Substitute back, giving 721/2 or (21/2)1/7.
Step 4: Use the properties of exponents: (am)n=am⋅n.
Step 5: Calculate the exponent: (1/2)⋅(1/7)=1/14.
Step 6: This gives us 21/14, which is equal to 142.
Thus, the simplified expression is 142.
Therefore, the solution to the problem is 142.
Answer
142
Exercise #10
Solve the following exercise:
3729=
Video Solution
Step-by-Step Solution
To solve this problem, we need to evaluate 3729.
Step 1: Convert the expression into exponent form. We know 729=72921. Thus, 3729=(72921)31.
Step 2: Apply the formula for exponents (am)n=am⋅n. Thus, (72921)31=7292⋅1/31=72961.
Step 3: Find the base of 729 as a power of an integer. Observing, 729 = 36 because 36=729.
Step 4: Substitute the power of the base: 72961=(36)61=36⋅61=31=3.
Therefore, the solution to the problem is 3729=3.
Answer
3
Question 1
Solve the following exercise:
\( \sqrt[3]{\sqrt[3]{512}}= \)
Incorrect
Correct Answer:
2
Question 2
Solve the following exercise:
\( \sqrt[5]{\sqrt[]{1024}}= \)
Incorrect
Correct Answer:
2
Question 3
Solve the following exercise:
\( \sqrt{\sqrt{625}}= \)
Incorrect
Correct Answer:
5
Question 4
Solve the following exercise:
\( \sqrt[3]{\sqrt{64}}= \)
Incorrect
Correct Answer:
2
Question 5
Solve the following exercise:
\( \sqrt[3]{\sqrt{144}}= \)
Incorrect
Correct Answer:
\( \sqrt[6]{144} \)
Exercise #11
Solve the following exercise:
33512=
Video Solution
Step-by-Step Solution
To solve this problem, we'll proceed with the following steps:
Step 1: Compute 3512.
Given 512, recognize that 512=29 since 29=512. Thus, we have:
3512=329=29/3=23=8.
Step 2: Compute 38.
From Step 1, we found 3512=8. Now find 38:
38=323=23/3=2.
Therefore, the solution to the expression 33512 is 2.
Answer
2
Exercise #12
Solve the following exercise:
51024=
Video Solution
Step-by-Step Solution
To solve this problem, we'll follow these steps:
Step 1: Simplify the inner square root.
Step 2: Apply the formula for nested roots.
Step 3: Compute the fifth root.
Now, let's work through each step:
Step 1: Calculate the inner square root.
We have 1024. We know that 1024=210, so 1024=210.
Applying the property of roots, 210=2210=25=32.
Step 2: Now, apply the fifth root to the result from step 1.
We need to find 532.
Step 3: Simplify using the properties of exponents.
From 532=525, we have 255=21=2.
Therefore, the solution to the problem is 2.
Answer
2
Exercise #13
Solve the following exercise:
625=
Video Solution
Step-by-Step Solution
To solve this problem, we'll follow these steps:
Step 1: Evaluate the innermost square root, 625.
Step 2: Evaluate the square root of the result from Step 1.
Now, let's work through each step:
Step 1: Evaluate 625.
The square root of 625 is 25, since 25×25=625. Thus, 625=25.
Step 2: Evaluate 25.
The square root of 25 is 5, since 5×5=25. Thus, 25=5.
Therefore, the solution to the problem 625 is 5.
Answer
5
Exercise #14
Solve the following exercise:
364=
Video Solution
Step-by-Step Solution
To solve this problem, we'll express the nested roots using exponents and then simplify:
We start with the inner expression:
64 is equivalent to 641/2.
Next, apply the cube root:
3641/2 is equivalent to (641/2)1/3.
Using properties of exponents, we simplify the expression:
(641/2)1/3=64(1/2)⋅(1/3)=641/6.
Now, evaluate 641/6:
Since 64=26, we have:
641/6=(26)1/6=26⋅(1/6)=21=2.
Therefore, the solution to the exercise 364 is 2.
Answer
2
Exercise #15
Solve the following exercise:
3144=
Video Solution
Step-by-Step Solution
To solve this problem, let's follow these steps:
Express the square root as a fractional exponent.
Express the cube root as another fractional exponent.
Multiply the exponents together using the rule (am)n=am×n.
Recapture the result as a root expression.
Let's apply these steps:
Step 1: The square root of 144 can be expressed as 1441/2.
Step 2: We need the cube root of this expression, so we have (1441/2)1/3.
Step 3: Using the property of exponents (am)n=am×n, we multiply the exponents: (1441/2)1/3=144(1/2)×(1/3)=1441/6.
Step 4: Re-express this as a root: Since 1441/6 is equivalent to the sixth root, we have 6144.
Therefore, the solution to the problem is 6144, which corresponds to choice 3.