Simplify 5/( square root of 17)

Solution:

Step 1:

Rationalize the denominator of $\frac{5}{\sqrt{17}}$ by multiplying both numerator and denominator by $\sqrt{17}$: $\frac{5}{\sqrt{17}}\cdot \frac{\sqrt{17}}{\sqrt{17}}$.

Step 2:

Proceed to simplify the expression.

Step 2.1:

Multiply the numerator by $\sqrt{17}$: $\frac{5\sqrt{17}}{\sqrt{17}\cdot \sqrt{17}}$.

Step 2.2:

Express $\sqrt{17}$ as ${\left(\sqrt{17}\right)}^1$: $\frac{5\sqrt{17}}{{\left(\sqrt{17}\right)}^1\cdot \sqrt{17}}$.

Step 2.3:

Again, express the second $\sqrt{17}$ as ${\left(\sqrt{17}\right)}^1$: $\frac{5\sqrt{17}}{{\left(\sqrt{17}\right)}^1\cdot {\left(\sqrt{17}\right)}^1}$.

Step 2.4:

Apply the exponent multiplication rule $a^m\cdot a^n=a^{m+n}$: $\frac{5\sqrt{17}}{{\left(\sqrt{17}\right)}^{1+1}}$.

Step 2.5:

Add the exponents: $\frac{5\sqrt{17}}{{\left(\sqrt{17}\right)}^2}$.

Step 2.6:

Convert ${\left(\sqrt{17}\right)}^2$ to $17$.

Step 2.6.1:

Represent $\sqrt{17}$ as ${\left(17\right)}^{\frac{1}{2}}$: $\frac{5\sqrt{17}}{{\left({\left(17\right)}^{\frac{1}{2}}\right)}^2}$.

Step 2.6.2:

Use the power of a power rule ${\left(a^m\right)}^n=a^{mn}$: $\frac{5\sqrt{17}}{{\left(17\right)}^{\frac{1}{2}\cdot 2}}$.

Step 2.6.3:

Multiply the exponents: $\frac{5\sqrt{17}}{{\left(17\right)}^{\frac{2}{2}}}$.

Step 2.6.4:

Simplify the exponent by cancelling out the common factor of 2.

Step 2.6.4.1:

Cancel out the common factor: $\frac{5\sqrt{17}}{{\left(17\right)}^{\frac{\cancel{2}}{\cancel{2}}}}$.

Step 2.6.4.2:

Rewrite the expression: $\frac{5\sqrt{17}}{17}$.

Step 2.6.5:

Finalize the simplification: $\frac{5\sqrt{17}}{17}$.

Step 3:

Present the final result in different forms.

Exact Form: $\frac{5\sqrt{17}}{17}$ Decimal Form: Approximately $1.21267812\dots$

Knowledge Notes:

1. Rationalizing the Denominator: This process involves removing the radical from the denominator of a fraction by multiplying both the numerator and the denominator by an appropriate form of 1 that contains the radical.

2. Simplifying Radicals: When simplifying radicals, it's important to combine like terms and apply the rules of exponents correctly.

3. Exponent Rules: The power rule states that $a^m\cdot a^n=a^{m+n}$, and the power of a power rule states that $(a^m{)}^n=a^{mn}$.

4. Square Roots: The square root of a number $x$ is represented as $\sqrt{x}$, and it can also be written as $x^{\frac{1}{2}}$.

5. Decimal Approximation: The exact form of a radical expression can be approximated to a decimal using a calculator or other computational tools. However, the exact form is often preferred in mathematics to maintain precision.