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:
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 #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:
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 #4
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 #5
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
Question 1
Solve the following exercise:
\( \sqrt{\sqrt{2}}= \)
Incorrect
Correct Answer:
\( \sqrt[4]{2} \)
Question 2
Solve the following exercise:
\( \sqrt{\sqrt{4}}= \)
Incorrect
Correct Answer:
\( \sqrt{2} \)
Question 3
Solve the following exercise:
\( \sqrt[]{\sqrt{8}}= \)
Incorrect
Correct Answer:
\( 8^{\frac{1}{4}} \)
Question 4
Complete the following exercise:
\( \sqrt[3]{\sqrt{36}}= \)
Incorrect
Correct Answer:
\( 36^{\frac{1}{6}} \)
Question 5
Complete the following exercise:
\( \sqrt[3]{\sqrt{64\cdot x^{12}}=} \)
Incorrect
Correct Answer:
\( 2x^2 \)
Exercise #6
Solve the following exercise:
2=
Video Solution
Step-by-Step Solution
To solve 2, we will use the property of roots.
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 #7
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 #8
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
Exercise #9
Complete the following exercise:
336=
Video Solution
Step-by-Step Solution
To solve this problem, let's analyze and simplify the given expression 336.
Step 1: Identify the root operations. We have a square root, 36, and a cube root, 3….
Step 2: Use the formula for roots for a root of a root: mnx=xmn1.
Step 3: Apply this formula to the problem. In this case, the first operation is a square root, which can be written as 3621, and the second operation is a cube root. Therefore, 336=(3621)31.
Step 4: Simplify using the power of a power rule, which allows us to multiply exponents: (3621)31=362×31=3661.
Thus, the expression 336 simplifies to 3661.
Answer
3661
Exercise #10
Complete the following exercise:
364⋅x12=
Video Solution
Step-by-Step Solution
To solve the problem 364⋅x12, follow these detailed steps:
Step 1: Simplify the inner expression. The expression inside the radical is 64⋅x12.
Step 2: Simplify the inner square root.
First, we need to find 64⋅x12.
The square root of a product can be expressed as the product of the square roots: 64⋅x12.
Simplifying further, we find:
64=8, since 82=64.
x12=x6, because (x6)2=x12.
Thus, the inner square root becomes 8x6.
Step 3: Simplify using the cube root.
Next, apply the cube root to the result of the inner square root: 38x6.
The cube root of a product can also be expressed as the product of the cube roots:
38=2, since 23=8.
3x6=x6/3=x2, because (x2)3=x6.
Thus, the expression simplifies to 2x2.
Therefore, the solution to this problem is 2x2, which corresponds to choice 2 in the provided options.
Answer
2x2
Question 1
Complete the following exercise:
\( \sqrt[8]{\sqrt{x^8}}= \)
Incorrect
Correct Answer:
\( \sqrt{x} \)
Question 2
Complete the following exercise:
\( \sqrt{\sqrt{16\cdot x^2}}= \)
Incorrect
Correct Answer:
\( 2\sqrt{x} \)
Question 3
Complete the following exercise:
\( \sqrt{\sqrt{3x^2}}= \)
Incorrect
Correct Answer:
\( \sqrt[4]{3}\cdot\sqrt{x} \)
Question 4
Complete the following exercise:
\( \sqrt[]{\sqrt{5x^4}}= \)
Incorrect
Correct Answer:
\( \sqrt[4]{5}\cdot x \)
Question 5
Complete the following exercise:
\( \sqrt{\sqrt{81\cdot x^4}}= \)
Incorrect
Correct Answer:
\( 3x \)
Exercise #11
Complete the following exercise:
8x8=
Video Solution
Step-by-Step Solution
To solve the problem 8x8, we'll simplify the expression using exponent rules:
Step 1: Express the inner square root using exponents. We know x8=(x8)1/2=x8⋅1/2=x4.
Step 2: Express the entire expression with the 8th root as an exponent. We have 8x4=(x4)1/8.
Step 3: Simplify the expression, using (xa)b=xa⋅b. Therefore, (x4)1/8=x4⋅1/8=x1/2.
Step 4: Recognize x1/2 is another way to write x.
Thus, the expression simplifies to x.
Answer
x
Exercise #12
Complete the following exercise:
16⋅x2=
Video Solution
Step-by-Step Solution
To solve the expression 16⋅x2, follow these steps:
Step 2: Calculate each component:
- 16=4 because 161/2=4.
- x2=x assuming x is non-negative.
Step 3: Combine results from Step 2: 16⋅x2=4⋅x=4x.
Step 4: Simplify the outer square root: 4x.
Applying 4x=4⋅x, we have 4=2.
Thus, 4x=2⋅x=2x.
Therefore, the simplified form of 16⋅x2 is 2x. This corresponds to choice 1.
Answer
2x
Exercise #13
Complete the following exercise:
3x2=
Video Solution
Step-by-Step Solution
To solve 3x2, follow these steps:
Step 1: Express the problem using exponentiation. The expression 3x2 can be written as (3x2)21.
Step 2: Take the square root of the first expression. This can be expressed as ((3x2)21)21.
Step 3: Use the property (am)n=am⋅n. Thus, ((3x2)21)21=(3x2)41.
Step 4: Simplify further using exponent rules: (3x2)41 becomes (341⋅(x2)41), which simplifies to 43⋅x21.
Step 5: Recognize this as 43⋅x, since x21 is x.
Therefore, the simplified form of the given expression is 43⋅x.
Answer
43⋅x
Exercise #14
Complete the following exercise:
5x4=
Video Solution
Step-by-Step Solution
To solve the expression 5x4, let's go step-by-step:
Step 1: Simplify the inner expression 5x4. Using the rule for square roots, we can rewrite 5x4 as (5x4)1/2. This expression can be further simplified to 51/2⋅(x4)1/2=5⋅x2.
Step 2: Take the square root of the simplified expression. This means we apply another square root to 5⋅x2, resulting in (5⋅x2)1/2=(5)1/2⋅(x2)1/2.
Step 3: Simplify each component: 45⋅x. We find that (5)1/2 simplifies to 45 and (x2)1/2 to x.
Therefore, the simplified expression is 45⋅x.
Answer
45⋅x
Exercise #15
Complete the following exercise:
81⋅x4=
Video Solution
Step-by-Step Solution
To solve the problem 81⋅x4, we need to simplify this expression using properties of exponents and square roots.
Step 1: Simplify the inner square root
The expression inside the first square root is 81⋅x4. We can rewrite this using exponents: 81=92 and x4=(x2)2. Thus, 81⋅x4=(9x2)2.
Step 2: Apply the inner square root
Taking the square root of (9x2)2 gives us: (9x2)2=9x2, because a2=a where a is a non-negative real number.
Step 3: Simplify the outer square root
Now, we take the square root of the result from the inner root: 9x2=9⋅x2=3⋅x=3x, since x2=x given x is non-negative.