Simplify 5/( square root of 17)
The question requests to express the fraction 5/(√17), which involves a number divided by the square root of another number, in a simplified form. This typically means rationalizing the denominator so that there are no square roots in the denominator of the fraction. Rationalizing the denominator often involves multiplying both the numerator and denominator by the square root that is in the denominator, in order to eliminate the square root from the denominator.
$\frac{5}{\sqrt{17}}$
Answer
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 $1.21267812 \ldots$
Knowledge Notes:
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.
Simplifying Radicals: When simplifying radicals, it's important to combine like terms and apply the rules of exponents correctly.
Exponent Rules: The power rule states that $a^m \cdot a^n = a^{m+n}$, and the power of a power rule states that $(a^m)^n = a^{mn}$.
Square Roots: The square root of a number $x$ is represented as $\sqrt{x}$, and it can also be written as $x^{\frac{1}{2}}$.
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.
