Root of a Root Practice Problems & Solutions

Master nested radicals with step-by-step root of root practice problems. Learn to simplify radical expressions using multiplication of indices.

πŸ“šWhat You'll Master in This Practice Session
  • Apply the root of root rule by multiplying indices together
  • Simplify nested radical expressions like βˆ›βˆš64 into single radicals
  • Convert compound root expressions using index multiplication properties
  • Solve multi-step problems involving nested square roots and cube roots
  • Recognize when to apply root of root simplification in complex expressions
  • Build confidence with progressive difficulty levels from basic to advanced

Understanding Root of a Root

Complete explanation with examples

Square Roots

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:Β Β 

root of root unknowns

Detailed explanation

Practice Root of a Root

Test your knowledge with 13 quizzes

Solve the following exercise:

\( \sqrt{\sqrt{625}}= \)

Examples with solutions for Root of a Root

Step-by-step solutions included
Exercise #1

Solve the following exercise:

2= \sqrt{\sqrt{2}}=

Step-by-Step Solution

To solve 2\sqrt{\sqrt{2}}, we will use the property of roots.

  • Step 1: Recognize that 2\sqrt{\sqrt{2}} involves two square roots.
  • Step 2: Each square root can be expressed using exponents: 2=21/2\sqrt{2} = 2^{1/2}.
  • Step 3: Therefore, 2=(21/2)1/2\sqrt{\sqrt{2}} = (2^{1/2})^{1/2}.
  • Step 4: Apply the formula for the root of a root: (xa)b=xab(x^{a})^{b} = x^{ab}.
  • Step 5: For (21/2)1/2(2^{1/2})^{1/2}, this means we compute the product of the exponents: (1/2)Γ—(1/2)=1/4(1/2) \times (1/2) = 1/4.
  • Step 6: The expression simplifies to 21/42^{1/4}, which is written as 24\sqrt[4]{2}.

Therefore, 2=24\sqrt{\sqrt{2}} = \sqrt[4]{2}.

This corresponds to choice 2: 24 \sqrt[4]{2} .

The solution to the problem is 24 \sqrt[4]{2} .

Answer:

24 \sqrt[4]{2}

Video Solution
Exercise #2

Solve the following exercise:

535= \sqrt[5]{\sqrt[3]{5}}=

Step-by-Step Solution

To solve the problem of finding 535 \sqrt[5]{\sqrt[3]{5}} , 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: 53=51/3 \sqrt[3]{5} = 5^{1/3} .
  • Step 2: Apply the outer root.
    The square root to the fifth power is expressed as: 51/35=(51/3)1/5 \sqrt[5]{5^{1/3}} = (5^{1/3})^{1/5} .
  • Step 3: Combine the exponents.
    Using the exponent rule (am)n=amΓ—n(a^m)^n = a^{m \times n}, we get:
    (51/3)1/5=5(1/3)Γ—(1/5)=51/15(5^{1/3})^{1/5} = 5^{(1/3) \times (1/5)} = 5^{1/15}.
  • Step 4: Convert the exponent back to root form.
    This can be written as 515 \sqrt[15]{5} .

Therefore, the simplified expression of 535 \sqrt[5]{\sqrt[3]{5}} is 515 \sqrt[15]{5} .

Answer:

515 \sqrt[15]{5}

Video Solution
Exercise #3

Solve the following exercise:

11010= \sqrt[10]{\sqrt[10]{1}}=

Step-by-Step Solution

To solve this problem, we'll observe the following process:

  • Step 1: Recognize the expression 11010 \sqrt[10]{\sqrt[10]{1}} involves nested roots.
  • Step 2: Apply the formula for nested roots: xmn=xnβ‹…m \sqrt[n]{\sqrt[m]{x}} = \sqrt[n \cdot m]{x} .
  • Step 3: Set n=10 n = 10 and m=10 m = 10 , resulting in 110Γ—10=1100 \sqrt[10 \times 10]{1} = \sqrt[100]{1} .
  • Step 4: Simplify 1100 \sqrt[100]{1} . Any root of 1 is 1, as 1k=1 1^k = 1 for any positive rational number k k .

Thus, the evaluation of the original expression 11010 \sqrt[10]{\sqrt[10]{1}} equals 1.

Comparing this result to the provided choices:

  • Choice 1 is 1 1 .
  • Choice 2 is 1100 \sqrt[100]{1} , which is also 1.
  • Choice 3 is 1=1 \sqrt{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.

Video Solution
Exercise #4

Solve the following exercise:

26= \sqrt[6]{\sqrt{2}}=

Step-by-Step Solution

Express the definition of root as a power:

an=a1n \sqrt[n]{a}=a^{\frac{1}{n}}

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 n=2

Meaning:

a=a2=a12 \sqrt{a}=\sqrt[2]{a}=a^{\frac{1}{2}}

Now convert the roots in the problem using the root definition provided above. :

26=2126=(212)16 \sqrt[6]{\sqrt{2}}=\sqrt[6]{2^{\frac{1}{2}}}=\big(2^{\frac{1}{2}}\big)^{\frac{1}{6}}

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=amβ‹…n (a^m)^n=a^{m\cdot n}

Apply this law to the expression that we obtained in the last stage:

(212)16=212β‹…16=21β‹…12β‹…6=2112 \big(2^{\frac{1}{2}}\big)^{\frac{1}{6}}=2^{\frac{1}{2}\cdot\frac{1}{6}}=2^{\frac{1\cdot1}{2\cdot6}}=2^{\frac{1}{12}}

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:

26=(212)16=2112 \sqrt[6]{\sqrt{2}}=\big(2^{\frac{1}{2}}\big)^{\frac{1}{6}} =2^{\frac{1}{12}}

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:

a1n=an a^{\frac{1}{n}} = \sqrt[n]{a}

Let's apply this law in order to present the expression we obtained in the last stage in root form:

2112=212 2^{\frac{1}{12}} =\sqrt[12]{2}

We obtain the following result: :

26=2112=212 \sqrt[6]{\sqrt{2}}=2^{\frac{1}{12}} =\sqrt[12]{2}

Therefore the correct answer is answer A.

Answer:

212 \sqrt[12]{2}

Video Solution
Exercise #5

Solve the following exercise:

8= \sqrt[]{\sqrt{8}}=

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\sqrt{8} = 8^{1/2}.

  • Step 2: Apply the root of a root property: 8=(8)1/2=(81/2)1/2\sqrt{\sqrt{8}} = (\sqrt{8})^{1/2} = (8^{1/2})^{1/2}.

  • Step 3: Simplify the expression using exponent rules: (81/2)1/2=8(1/2)β‹…(1/2)=81/4(8^{1/2})^{1/2} = 8^{(1/2) \cdot (1/2)} = 8^{1/4}.

The nested root expression simplifies to 81/48^{1/4}.

Therefore, the simplified expression of 8\sqrt{\sqrt{8}} is 814 8^{\frac{1}{4}} .

After comparing this result with the multiple choice answers, choice 2 is correct.

Answer:

814 8^{\frac{1}{4}}

Video Solution

Frequently Asked Questions

What is the root of a root rule in mathematics?

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The root of a root rule states that when you have a root applied to another root, you multiply the indices (orders) of both roots together. For example, βˆ›βˆšx becomes ⁢√x because 3 Γ— 2 = 6.

How do you simplify nested radicals step by step?

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To simplify nested radicals: 1) Identify the outer and inner root indices, 2) Multiply the indices together, 3) Apply the combined index to the radicand, 4) Simplify the final expression if possible.

Why does βˆ›βˆš64 equal 2?

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βˆ›βˆš64 simplifies to ⁢√64 using the root of root rule (3 Γ— 2 = 6). Since 2⁢ = 64, the sixth root of 64 equals 2.

What are common mistakes when solving root of root problems?

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Common mistakes include: adding indices instead of multiplying them, forgetting to apply the rule to the entire radicand, and not simplifying the final answer completely.

When do you use the root of root property in algebra?

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The root of root property is used when simplifying complex radical expressions, solving equations with nested radicals, and in advanced topics like fractional exponents and logarithms.

How does root of root relate to exponent rules?

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Root of root follows the same pattern as the power of a power rule for exponents. Just as (x^a)^b = x^(ab), nested roots multiply their indices: ⁿ√ᡐ√x = ⁿᡐ√x.

Can you have more than two nested roots?

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Yes, you can have multiple nested roots. The rule extends by multiplying all indices together. For example, ⁴√³√²√x = ²⁴√x because 4 Γ— 3 Γ— 2 = 24.

What grade level typically learns root of root problems?

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Root of root problems are typically introduced in Algebra 2 or Pre-Calculus (grades 10-11) after students have mastered basic radical operations and properties.

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