Simplify square root of 196/( square root of 2)

Solution:

Step 1:

Rationalize the denominator by multiplying the expression $\frac{\sqrt{196}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ to get $\sqrt{\frac{196}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}}}$.

Step 2:

Simplify the expression within the square root.

Step 2.1:

Multiply the numerators and denominators to get $\sqrt{\frac{196\cdot \sqrt{2}}{\sqrt{2}\cdot \sqrt{2}}}$.

Step 2.2:

Recognize that $\sqrt{2}$ raised to the first power is $\sqrt{2}$, so we have $\sqrt{\frac{196\cdot \sqrt{2}}{(\sqrt{2}{)}^1\cdot \sqrt{2}}}$.

Step 2.3:

Again, acknowledge that $\sqrt{2}$ raised to the first power is $\sqrt{2}$, resulting in $\sqrt{\frac{196\cdot \sqrt{2}}{(\sqrt{2}{)}^1\cdot (\sqrt{2}{)}^1}}$.

Step 2.4:

Apply the exponent rule $a^m\cdot a^n=a^{m+n}$ to get $\sqrt{\frac{196\cdot \sqrt{2}}{(\sqrt{2}{)}^{1+1}}}$.

Step 2.5:

Add the exponents to simplify to $\sqrt{\frac{196\cdot \sqrt{2}}{(\sqrt{2}{)}^2}}$.

Step 2.6:

Convert $(\sqrt{2}{)}^2$ to $2$.

Step 2.6.1:

Express $\sqrt{2}$ as $2^{\frac{1}{2}}$ to get $\sqrt{\frac{196\cdot \sqrt{2}}{(2^{\frac{1}{2}}{)}^2}}$.

Step 2.6.2:

Apply the power rule $(a^m{)}^n=a^{mn}$ to obtain $\sqrt{\frac{196\cdot \sqrt{2}}{2^{\frac{1}{2}\cdot 2}}}$.

Step 2.6.3:

Multiply the exponents to simplify to $\sqrt{\frac{196\cdot \sqrt{2}}{2^{\frac{2}{2}}}}$.

Step 2.6.4:

Reduce the fraction to get $\sqrt{\frac{196\cdot \sqrt{2}}{2^1}}$.

Step 2.6.5:

Evaluate the exponent to finalize the simplification to $\sqrt{\frac{196\cdot \sqrt{2}}{2}}$.

Step 3:

Reduce the fraction by eliminating common factors.

Step 3.1:

Extract the factor of $2$ from $196\cdot \sqrt{2}$ to get $\sqrt{\frac{2\cdot (98\cdot \sqrt{2})}{2}}$.

Step 3.2:

Cancel out the common factors.

Step 3.2.1:

Factor out the $2$ from the denominator to have $\sqrt{\frac{2\cdot (98\cdot \sqrt{2})}{2\cdot 1}}$.

Step 3.2.2:

Eliminate the common factor of $2$ to simplify to $\sqrt{\frac{\cancel{2}\cdot (98\cdot \sqrt{2})}{\cancel{2}\cdot 1}}$.

Step 3.2.3:

Rewrite the simplified expression as $\sqrt{\frac{98\cdot \sqrt{2}}{1}}$.

Step 3.2.4:

Divide $98\cdot \sqrt{2}$ by $1$ to get $\sqrt{98\cdot \sqrt{2}}$.

Step 4:

Express $98\cdot \sqrt{2}$ in terms of its prime factors.

Step 4.1:

Factor $49$ from $98$ to write $\sqrt{49\cdot 2\cdot \sqrt{2}}$.

Step 4.2:

Rewrite $49$ as $7^2$ to get $\sqrt{7^2\cdot 2\cdot \sqrt{2}}$.

Step 4.3:

Enclose the terms under the radical as $\sqrt{7^2\cdot (2\cdot \sqrt{2})}$.

Step 5:

Extract terms from under the square root where possible to get $7\cdot \sqrt{2\cdot \sqrt{2}}$.

Step 6:

Present the result in both exact and decimal forms.

Exact Form: $7\cdot \sqrt{2\cdot \sqrt{2}}$ Decimal Form: $11.77254981\dots$

Knowledge Notes:

1. Rationalizing the Denominator: This process involves multiplying the numerator and denominator by a conjugate or a suitable term to eliminate square roots or other irrational numbers from the denominator.

2. Simplifying Radicals: When simplifying expressions involving radicals, we often combine like terms and reduce fractions to their simplest form.

3. Exponent Rules: The power rule states that $a^m\cdot a^n=a^{m+n}$ and $(a^m{)}^n=a^{mn}$. These rules are used to simplify expressions with exponents.

4. Square Roots: The square root of a number $x$, denoted as $\sqrt{x}$, is a value that, when multiplied by itself, gives $x$. The square root of a perfect square is an integer.

5. Prime Factorization: Breaking down a number into its prime factors can sometimes simplify the process of taking square roots, as squares of prime factors can be easily extracted from under the radical.

6. Algebraic Manipulation: This involves various techniques such as factoring, expanding, and canceling common factors to simplify expressions.