Simplify 3 square root of square root of 64
The problem asks to perform a mathematical simplification on the expression given. Specifically, it requires us to simplify the expression which contains nested radicals: a cube root (which is denoted by the radical symbol with an index of 3) of the square root (denoted by the radical symbol without an index, implying an index of 2) of the number 64. The task involves simplifying the radical expressions to the simplest form possible by performing the appropriate root operations on the number 64.
$3 \sqrt{\sqrt{64}}$
Answer
Solution:
Simplification Process:
Step 1: Express $64$ as a power of $8$. $3 \sqrt{\sqrt{64}} = 3 \sqrt{\sqrt{8^2}}$
Step 2: Extract the square root of the perfect square. $3 \sqrt{8}$
Step 3: Decompose $8$ into its prime factors.
Step 3.1: Isolate the perfect square factor from $8$. $3 \sqrt{4 \cdot 2}$ Step 3.2: Express $4$ as a power of $2$. $3 \sqrt{2^2 \cdot 2}$
Step 4: Remove the perfect square from under the radical sign. $3 \cdot 2 \sqrt{2}$
Step 5: Perform the multiplication outside the radical. $6 \sqrt{2}$
Step 6: Present the final answer in various forms.
Exact Form: $6 \sqrt{2}$ Decimal Form: Approximately $8.48528137$
Knowledge Notes:
To simplify an expression involving nested radicals and integer coefficients, we follow these steps:
Rewriting Numbers as Powers: Express the number under the radical in terms of its prime factors or as a power of another number to simplify the extraction of roots.
Simplifying Nested Radicals: When dealing with nested radicals, like $\sqrt{\sqrt{x}}$, we can simplify by recognizing that $\sqrt{\sqrt{x}} = \sqrt{x^{1/2}} = x^{1/4}$.
Extracting Square Roots: If the number under the radical is a perfect square or has a perfect square factor, we can take the square root of that factor out from under the radical.
Multiplication Outside the Radical: After extracting terms from under the radical, we can multiply them by any coefficients outside the radical.
Prime Factorization: This is the process of breaking down a composite number into its prime factors. For example, $8 = 2^3$, and $4 = 2^2$. This is useful when simplifying radicals because it can reveal perfect square factors.
Radical Simplification: The square root of a product of terms can be simplified by taking the square root of each term separately, as in $\sqrt{a^2 \cdot b} = a \sqrt{b}$, provided $a$ and $b$ are positive real numbers.
Final Forms: The simplified expression can be presented in its exact form (with radicals) or as a decimal approximation. The exact form is more precise, while the decimal form provides an intuitive understanding of the number's magnitude.
