Simplify square root of 2 square root of 8
The problem is asking to simplify a mathematical expression that involves nested square roots. Specifically, the expression to be simplified is the square root of 2 times the square root of 8. Simplification here likely refers to combining the nested roots into a single square root or simplifying the expression to its most reduced form, where no further simplification is possible using basic rules of arithmetic on square roots.
$\sqrt{2 \sqrt{8}}$
Answer
Solution:
Step 1:
Express $8$ as $2^3$ by factoring out the powers of $2$.
Step 1.1:
Break down $8$ into $4 \times 2$ to simplify the expression. $\sqrt{2 \sqrt{4 \cdot 2}}$
Step 1.2:
Represent $4$ as $2^2$. $\sqrt{2 \sqrt{2^2 \cdot 2}}$
Step 2:
Extract the square root of the perfect square from under the radical sign. $\sqrt{2 \cdot 2 \sqrt{2}}$
Step 3:
Express the numbers as powers where applicable.
Step 3.1:
Combine the $2$ outside the radical with the square root of $4$. $\sqrt{4 \sqrt{2}}$
Step 3.2:
Convert $4$ back into $2^2$. $\sqrt{2^2 \sqrt{2}}$
Step 4:
Remove the square root of the perfect square from under the radical. $2 \sqrt{\sqrt{2}}$
Step 5:
Express the nested square root as a fourth root. $2 \sqrt[4]{2}$
Step 6:
Present the final result in its exact and approximate decimal forms.
Exact Form: $2 \sqrt[4]{2}$ Decimal Form: $2.37841423 \ldots$
Knowledge Notes:
The problem involves simplifying a nested radical expression, which is an expression containing a radical within another radical. The steps taken to simplify such an expression include:
Factorization: Breaking down a number into its prime factors or into a product of smaller numbers that are easier to work with under the radical sign.
Radical Properties: Utilizing the properties of radicals to simplify expressions, such as $\sqrt{a^2} = a$ for non-negative $a$, and $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$.
Exponent Rules: Applying the rules of exponents, such as $a^{m \cdot n} = (a^m)^n$, to rewrite expressions in a more manageable form.
Nested Radicals: Simplifying nested radicals by recognizing patterns like $\sqrt{\sqrt{a}} = \sqrt[4]{a}$, which is a result of the property that $\sqrt[n]{\sqrt[m]{a}} = \sqrt[n \cdot m]{a}$.
Exact and Decimal Forms: The final answer can be presented in exact form using radicals and exponents, or in decimal form by calculating the numerical approximation.
Understanding these concepts is crucial for simplifying radical expressions and solving problems involving nested radicals.
