Simplify square root of 202/17

Solution:

Simplification Process:

Step 1: Separate the square root of the fraction into the square root of the numerator and denominator: $\sqrt{\frac{202}{17}} = \frac{\sqrt{202}}{\sqrt{17}}$.

Step 2: Rationalize the denominator by multiplying the fraction by $\frac{\sqrt{17}}{\sqrt{17}}$: $\frac{\sqrt{202}}{\sqrt{17}} \cdot \frac{\sqrt{17}}{\sqrt{17}}$.

Step 3: Simplify the expression by combining the terms in the denominator:

Step 3.1: Multiply the square roots in the numerator and denominator: $\frac{\sqrt{202} \cdot \sqrt{17}}{\sqrt{17} \cdot \sqrt{17}}$.

Step 3.2: Express $\sqrt{17}$ as a power: $\frac{\sqrt{202} \cdot \sqrt{17}}{(\sqrt{17})^1 \cdot \sqrt{17}}$.

Step 3.3: Repeat the expression of $\sqrt{17}$ as a power: $\frac{\sqrt{202} \cdot \sqrt{17}}{(\sqrt{17})^1 \cdot (\sqrt{17})^1}$.

Step 3.4: Apply the exponent rule $a^m \cdot a^n = a^{m+n}$: $\frac{\sqrt{202} \cdot \sqrt{17}}{(\sqrt{17})^{1+1}}$.

Step 3.5: Add the exponents: $\frac{\sqrt{202} \cdot \sqrt{17}}{(\sqrt{17})^2}$.

Step 3.6: Convert the squared square root back to the base number:

Step 3.6.1: Rewrite $\sqrt{17}$ using exponent notation: $\frac{\sqrt{202} \cdot \sqrt{17}}{((17)^{\frac{1}{2}})^2}$.

Step 3.6.2: Apply the power of a power rule: $\frac{\sqrt{202} \cdot \sqrt{17}}{(17)^{\frac{1}{2} \cdot 2}}$.

Step 3.6.3: Simplify the exponent: $\frac{\sqrt{202} \cdot \sqrt{17}}{(17)^{\frac{2}{2}}}$.

Step 3.6.4: Reduce the fraction in the exponent:

Step 3.6.4.1: Cancel out the common factor: $\frac{\sqrt{202} \cdot \sqrt{17}}{(17)^{\frac{\cancel{2}}{\cancel{2}}}}$.

Step 3.6.4.2: Finalize the denominator: $\frac{\sqrt{202} \cdot \sqrt{17}}{(17)^1}$.

Step 3.6.5: Evaluate the exponent: $\frac{\sqrt{202} \cdot \sqrt{17}}{17}$.

Step 4: Combine the square roots in the numerator using the product rule: $\frac{\sqrt{202 \cdot 17}}{17}$.

Step 4.2: Calculate the product under the square root: $\frac{\sqrt{3434}}{17}$.

Step 5: Present the result in its exact and decimal forms:

Exact Form: $\frac{\sqrt{3434}}{17}$ Decimal Form: $3.44707889 \ldots$

Knowledge Notes:

To simplify a square root of a fraction, you can follow these steps:

1. Separate the Square Roots: The square root of a fraction can be expressed as the square root of the numerator divided by the square root of the denominator.

2. Rationalize the Denominator: To avoid having a square root in the denominator, you can multiply the fraction by a form of 1 that contains the square root of the denominator in both the numerator and the denominator.

3. Simplify the Expression: Use algebraic rules to simplify the expression, such as combining like terms and using exponent rules.

Relevant exponent rules include:

• Power Rule: $a^m \cdot a^n = a^{m+n}$, which allows you to add exponents when multiplying like bases.

• Power of a Power Rule: $(a^m)^n = a^{m \cdot n}$, which allows you to multiply exponents when raising a power to another power.

When simplifying square roots:

• Product Rule for Radicals: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$, which allows you to combine square roots by multiplying the radicands (the numbers under the square roots).

Finally, the square of a square root, such as $(\sqrt{a})^2$, simplifies to the original number $a$ because the square root and the square are inverse operations.