Examples with solutions for Root of a Root: Using variables

Exercise #1

Complete the following exercise:

3x2= \sqrt{\sqrt{3x^2}}=

Video Solution

Step-by-Step Solution

To solve 3x2\sqrt{\sqrt{3x^2}}, follow these steps:

  • Step 1: Express the problem using exponentiation. The expression 3x2\sqrt{3x^2} can be written as (3x2)12(3x^2)^{\frac{1}{2}}.
  • Step 2: Take the square root of the first expression. This can be expressed as ((3x2)12)12((3x^2)^{\frac{1}{2}})^{\frac{1}{2}}.
  • Step 3: Use the property (am)n=amn(a^m)^n = a^{m \cdot n}. Thus, ((3x2)12)12=(3x2)14((3x^2)^{\frac{1}{2}})^{\frac{1}{2}} = (3x^2)^{\frac{1}{4}}.
  • Step 4: Simplify further using exponent rules: (3x2)14(3x^2)^{\frac{1}{4}} becomes (314(x2)14)(3^{\frac{1}{4}} \cdot (x^2)^{\frac{1}{4}}), which simplifies to 34x12\sqrt[4]{3} \cdot x^{\frac{1}{2}}.
  • Step 5: Recognize this as 34x\sqrt[4]{3} \cdot \sqrt{x}, since x12x^{\frac{1}{2}} is x\sqrt{x}.

Therefore, the simplified form of the given expression is 34x \sqrt[4]{3} \cdot \sqrt{x} .

Answer

34x \sqrt[4]{3}\cdot\sqrt{x}

Exercise #2

Complete the following exercise:

16x2= \sqrt{\sqrt{16\cdot x^2}}=

Video Solution

Step-by-Step Solution

To solve the expression 16x2 \sqrt{\sqrt{16 \cdot x^2}} , follow these steps:

  • Step 1: Simplify the innermost root 16x2 \sqrt{16 \cdot x^2} .
    Here, apply ab=ab \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} . Thus, 16x2=16x2 \sqrt{16 \cdot x^2} = \sqrt{16} \cdot \sqrt{x^2} .
  • Step 2: Calculate each component:
    - 16=4 \sqrt{16} = 4 because 161/2=4 16^{1/2} = 4 .
    - x2=x \sqrt{x^2} = x assuming x x is non-negative.
  • Step 3: Combine results from Step 2: 16x2=4x=4x \sqrt{16} \cdot \sqrt{x^2} = 4 \cdot x = 4x .
  • Step 4: Simplify the outer square root: 4x \sqrt{4x} .
    Applying 4x=4x \sqrt{4x} = \sqrt{4} \cdot \sqrt{x} , we have 4=2 \sqrt{4} = 2 .
    Thus, 4x=2x=2x \sqrt{4x} = 2 \cdot \sqrt{x} = 2\sqrt{x} .

Therefore, the simplified form of 16x2 \sqrt{\sqrt{16 \cdot x^2}} is 2x 2\sqrt{x} . This corresponds to choice 1.

Answer

2x 2\sqrt{x}

Exercise #3

Complete the following exercise:

81x4= \sqrt{\sqrt{81\cdot x^4}}=

Video Solution

Step-by-Step Solution

To solve the problem 81x4 \sqrt{\sqrt{81 \cdot x^4}} , 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 81x4 81 \cdot x^4 . We can rewrite this using exponents:
    81=92 81 = 9^2 and x4=(x2)2 x^4 = (x^2)^2 . Thus, 81x4=(9x2)2 81 \cdot x^4 = (9x^2)^2 .
  • Step 2: Apply the inner square root
    Taking the square root of (9x2)2 (9x^2)^2 gives us:
    (9x2)2=9x2 \sqrt{(9x^2)^2} = 9x^2 , because a2=a \sqrt{a^2} = a where a 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=9x2=3x=3x \sqrt{9x^2} = \sqrt{9} \cdot \sqrt{x^2} = 3 \cdot x = 3x , since x2=x \sqrt{x^2} = x given x x is non-negative.

Therefore, the solution to the problem is 3x 3x .

Answer

3x 3x

Exercise #4

Complete the following exercise:

64x12=3 \sqrt[3]{\sqrt{64\cdot x^{12}}=}

Video Solution

Step-by-Step Solution

To solve the problem 64x123 \sqrt[3]{\sqrt{64 \cdot x^{12}}} , follow these detailed steps:

  • Step 1: Simplify the inner expression.
    The expression inside the radical is 64x12 64 \cdot x^{12} .
  • Step 2: Simplify the inner square root.

    First, we need to find 64x12 \sqrt{64 \cdot x^{12}} .

    The square root of a product can be expressed as the product of the square roots: 64x12 \sqrt{64} \cdot \sqrt{x^{12}} .

    Simplifying further, we find:

    • 64=8 \sqrt{64} = 8 , since 82=64 8^2 = 64 .
    • x12=x6 \sqrt{x^{12}} = x^{6} , because (x6)2=x12 (x^{6})^2 = x^{12} .

    Thus, the inner square root becomes 8x6 8x^6 .

  • Step 3: Simplify using the cube root.

    Next, apply the cube root to the result of the inner square root: 8x63 \sqrt[3]{8x^6} .

    The cube root of a product can also be expressed as the product of the cube roots:

    • 83=2 \sqrt[3]{8} = 2 , since 23=8 2^3 = 8 .
    • x63=x6/3=x2 \sqrt[3]{x^6} = x^{6/3} = x^{2} , because (x2)3=x6 (x^2)^3 = x^6 .

    Thus, the expression simplifies to 2x2 2x^2 .

Therefore, the solution to this problem is 2x2 2x^2 , which corresponds to choice 2 in the provided options.

Answer

2x2 2x^2

Exercise #5

Complete the following exercise:

x88= \sqrt[8]{\sqrt{x^8}}=

Video Solution

Step-by-Step Solution

To solve the problem x88 \sqrt[8]{\sqrt{x^8}} , we'll simplify the expression using exponent rules:

  • Step 1: Express the inner square root using exponents. We know x8=(x8)1/2=x81/2=x4 \sqrt{x^8} = (x^8)^{1/2} = x^{8 \cdot 1/2} = x^4 .
  • Step 2: Express the entire expression with the 8th root as an exponent. We have x48=(x4)1/8 \sqrt[8]{x^4} = (x^4)^{1/8} .
  • Step 3: Simplify the expression, using (xa)b=xab (x^a)^{b} = x^{a \cdot b} . Therefore, (x4)1/8=x41/8=x1/2 (x^4)^{1/8} = x^{4 \cdot 1/8} = x^{1/2} .
  • Step 4: Recognize x1/2 x^{1/2} is another way to write x \sqrt{x} .

Thus, the expression simplifies to x \sqrt{x} .

Answer

x \sqrt{x}

Exercise #6

Complete the following exercise:

5x4= \sqrt[]{\sqrt{5x^4}}=

Video Solution

Step-by-Step Solution

To solve the expression 5x4\sqrt[]{\sqrt{5x^4}}, let's go step-by-step:

  • Step 1: Simplify the inner expression 5x4\sqrt{5x^4}. Using the rule for square roots, we can rewrite 5x4\sqrt{5x^4} as (5x4)1/2(5x^4)^{1/2}. This expression can be further simplified to 51/2(x4)1/2=5x25^{1/2} \cdot (x^4)^{1/2} = \sqrt{5} \cdot x^2.
  • Step 2: Take the square root of the simplified expression. This means we apply another square root to 5x2\sqrt{5} \cdot x^2, resulting in (5x2)1/2=(5)1/2(x2)1/2(\sqrt{5} \cdot x^2)^{1/2} = (\sqrt{5})^{1/2} \cdot (x^2)^{1/2}.
  • Step 3: Simplify each component: 54x\sqrt[4]{5} \cdot x. We find that (5)1/2(\sqrt{5})^{1/2} simplifies to 54\sqrt[4]{5} and (x2)1/2(x^2)^{1/2} to xx.

Therefore, the simplified expression is 54x \sqrt[4]{5} \cdot x .

Answer

54x \sqrt[4]{5}\cdot x

Exercise #7

Comlete the following exercise:

x2010= \sqrt[10]{\sqrt{x^{20}}}=

Video Solution

Step-by-Step Solution

To solve the problem, we'll simplify x2010 \sqrt[10]{\sqrt{x^{20}}} using properties of exponents and roots:

  • Step 1: Convert each part into exponent form. We have x20 \sqrt{x^{20}} , which can be rewritten as (x20)1/2(x^{20})^{1/2}.
  • Step 2: Simplify the expression (x20)1/2(x^{20})^{1/2}. Using the power of a power property, this becomes x20(1/2)=x10x^{20 \cdot (1/2)} = x^{10}.
  • Step 3: Apply the 10th root to the simplified result: x1010\sqrt[10]{x^{10}}, which can be written as (x10)1/10(x^{10})^{1/10}.
  • Step 4: Simplify (x10)1/10(x^{10})^{1/10} using the power of a power property again, we get x(10(1/10))=x1=xx^{(10 \cdot (1/10))} = x^1 = x.

Therefore, the expression simplifies to x x .

Conclusion: The solution to the problem is x x .

Answer

x x

Exercise #8

Complete the following exercise:

x205= \sqrt[5]{\sqrt{x^{20}}}=

Video Solution

Step-by-Step Solution

To solve the problem x205 \sqrt[5]{\sqrt{x^{20}}} , we will convert the roots into fractional exponents and simplify:

The expression x20 \sqrt{x^{20}} can be represented as:

(x20)12=x2012=x10 (x^{20})^{\frac{1}{2}} = x^{20 \cdot \frac{1}{2}} = x^{10}

Now, taking the fifth root of x10 x^{10} , we have:

(x10)15=x1015=x2 (x^{10})^{\frac{1}{5}} = x^{10 \cdot \frac{1}{5}} = x^{2}

Therefore, the original expression x205 \sqrt[5]{\sqrt{x^{20}}} simplifies to x2 x^2 .

Answer

x2 x^2

Exercise #9

Complete the following exercise:

x126= \sqrt[6]{\sqrt{x^{12}}}=

Video Solution

Step-by-Step Solution

To solve x126\sqrt[6]{\sqrt{x^{12}}}, we will follow these steps:

  • Step 1: Simplify the inner radical expression x12\sqrt{x^{12}}.
  • Step 2: Use the property of roots, expressing x12\sqrt{x^{12}} as a power of xx.
  • Step 3: Use the result from step 1 in the outer root expression.
  • Step 4: Simplify the entire expression using exponent rules.

Now, let's perform each of these steps:

Step 1: Simplify x12\sqrt{x^{12}}.
x12=x12/2=x6\sqrt{x^{12}} = x^{12/2} = x^6.

Step 2: Simplify the outer expression x66\sqrt[6]{x^6}.
x66=(x6)1/6\sqrt[6]{x^6} = (x^6)^{1/6}.

Step 3: Apply the exponent rule (am)n=am×n(a^m)^{n} = a^{m \times n}.
(x6)1/6=x6×1/6=x1=x(x^6)^{1/6} = x^{6 \times 1/6} = x^1 = x.

Therefore, the simplified expression is x\boxed{x}.

Thus, the solution to x126\sqrt[6]{\sqrt{x^{12}}} is xx.

Answer

x x

Exercise #10

Complete the following exercise:

25x66 \sqrt[6]{\sqrt{25x^6}}

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Convert the expression under the roots to fractional exponents.
  • Step 2: Simplify the fractional exponents.
  • Step 3: Return the expression to root form if needed.

Now, let's work through each step:

Step 1: We have the expression 25x66 \sqrt[6]{\sqrt{25x^6}} . Begin by simplifying the inner square root:

25x6=(25x6)1/2=251/2(x6)1/2\sqrt{25x^6} = (25x^6)^{1/2} = 25^{1/2} \cdot (x^6)^{1/2}.

We know 251/2=5 25^{1/2} = 5 and (x6)1/2=x61/2=x3(x^6)^{1/2} = x^{6 \cdot 1/2} = x^3.

So, 25x6=5x3 \sqrt{25x^6} = 5x^3 .

Step 2: Apply the outer sixth root:

5x36=(5x3)1/6=51/6(x3)1/6=51/6x1/2\sqrt[6]{5x^3} = (5x^3)^{1/6} = 5^{1/6} \cdot (x^3)^{1/6} = 5^{1/6} \cdot x^{1/2}.

Convert 51/6 5^{1/6} as 2512 \sqrt[12]{25} :

Since 51/6=(51/2)1/3=51/3=2512 5^{1/6} = (5^{1/2})^{1/3} = \sqrt{5}^{1/3} = \sqrt[12]{25} ,

we have 51/6=2512 5^{1/6} = \sqrt[12]{25} .

Therefore, the solution becomes:

x2512 \sqrt{x} \cdot \sqrt[12]{25} .

Therefore, the solution to the problem is x2512 \sqrt{x}\cdot\sqrt[12]{25} .

Answer

x2512 \sqrt{x}\cdot\sqrt[12]{25}

Exercise #11

Complete the following exercise:

49x34= \sqrt[4]{\sqrt[3]{49\cdot x}}=

Video Solution

Step-by-Step Solution

To simplify the given expression 49x34 \sqrt[4]{\sqrt[3]{49 \cdot x}} , we will use the property of roots as fractional exponents.

  • Step 1: Convert the cube root to a fractional exponent. We have 49x3=(49x)1/3 \sqrt[3]{49 \cdot x} = (49 \cdot x)^{1/3} .
  • Step 2: Apply the fourth root to the result, expressed as a fractional exponent. Thus, (49x)1/34=((49x)1/3)1/4 \sqrt[4]{(49 \cdot x)^{1/3}} = ((49 \cdot x)^{1/3})^{1/4} .
  • Step 3: Combine the exponents by multiplying the fractions: ((49x)1/3)1/4=(49x)(1/3)×(1/4)=(49x)1/12 ((49 \cdot x)^{1/3})^{1/4} = (49 \cdot x)^{(1/3) \times (1/4)} = (49 \cdot x)^{1/12} .
  • Step 4: Recognize that the result (49x)1/12 (49 \cdot x)^{1/12} can be expressed as 49x12 \sqrt[12]{49 \cdot x} .

Therefore, the solution to the problem is 49x12 \sqrt[12]{49x} .

Answer

49x12 \sqrt[12]{49x}

Exercise #12

Complete the following exercise:

144x33= \sqrt[3]{\sqrt{144x^3}}=

Video Solution

Step-by-Step Solution

To solve the expression 144x33 \sqrt[3]{\sqrt{144x^3}} , let's proceed step by step:

  • Step 1: Express the inner square root using fractional exponents:
    144x3=(144x3)1/2 \sqrt{144x^3} = (144x^3)^{1/2}
  • Step 2: Apply the cube root to the expression obtained in Step 1:
    (144x3)1/23=((144x3)1/2)1/3 \sqrt[3]{(144x^3)^{1/2}} = ((144x^3)^{1/2})^{1/3}
  • Step 3: Use the exponent rule (am)n=amn(a^m)^n = a^{m \cdot n}:
    ((144x3)1/2)1/3=(144x3)(1/2)(1/3)=(144x3)1/6 ((144x^3)^{1/2})^{1/3} = (144x^3)^{(1/2) \cdot (1/3)} = (144x^3)^{1/6}
  • Step 4: Separate the powers for 144 and x3 x^3 :
    (144)1/6(x3)1/6=1446x3/6=1446x1/2=1446x (144)^{1/6} \cdot (x^3)^{1/6} = \sqrt[6]{144} \cdot x^{3/6} = \sqrt[6]{144} \cdot x^{1/2} = \sqrt[6]{144} \cdot \sqrt{x}

Therefore, the simplified expression is 1446x \sqrt[6]{144} \cdot \sqrt{x} , which corresponds to the third choice.

Answer

1446x \sqrt[6]{144}\cdot\sqrt{x}

Exercise #13

Complete the following exercise:

512x2733= \sqrt[3]{\sqrt[3]{512x^{27}}}=

Video Solution

Step-by-Step Solution

To solve the given problem, we'll follow these steps:

  • Step 1: Simplify the innermost cube root 512x273\sqrt[3]{512x^{27}}.
  • Step 2: Simplify the next cube root ()3\sqrt[3]{(\cdot)} from the result of step 1.

Let's go through each step:

Step 1: Consider the expression 512x273\sqrt[3]{512x^{27}}.
First, evaluate 5123\sqrt[3]{512}. Since 512=83512 = 8^3, we have 5123=8\sqrt[3]{512} = 8.
For x273\sqrt[3]{x^{27}}, use the property amn=am/n\sqrt[n]{a^m} = a^{m/n}:
x273=x27/3=x9\sqrt[3]{x^{27}} = x^{27/3} = x^9.
Thus, 512x273=8x9\sqrt[3]{512x^{27}} = 8x^9.

Step 2: Now, evaluate the outer cube root 8x93\sqrt[3]{8x^9}.
83=2\sqrt[3]{8} = 2 since 8=238 = 2^3.
For x93\sqrt[3]{x^9}, again use the rule amn=am/n\sqrt[n]{a^m} = a^{m/n}:
x93=x9/3=x3\sqrt[3]{x^9} = x^{9/3} = x^3.
Therefore, 8x93=2x3\sqrt[3]{8x^9} = 2x^3.

In conclusion, the simplified expression is 2x32x^3.

Thus, the solution to the problem is 2x3 2x^3 , which corresponds to choice 3.

Answer

2x3 2x^3

Exercise #14

Solve the following exercise:

4x16= \sqrt{\sqrt{4x^{16}}}=

Video Solution

Answer

x244 x^2\cdot\sqrt[4]{4}

Exercise #15

Complete the following exercise:

100x4= \sqrt[4]{\sqrt{100x}}=

Video Solution

Answer

100x8 \sqrt[8]{100x}