Simplify (12 square root of 21-16)/( square root of 4)

Solution:

Step:1

Reduce the square root in the denominator.

Step:1.1

Express $4$ as $2^2$. $\frac{12\sqrt{21}-16}{\sqrt{2^2}}$

Step:1.2

Extract the square root of the perfect square. $\frac{12\sqrt{21}-16}{2}$

Step:2

Simplify by removing the common factor.

Step:2.1

Extract the factor of $2$ from $12\sqrt{21}$. $\frac{2(6\sqrt{21})-16}{2}$

Step:2.2

Extract the factor of $2$ from $-16$. $\frac{2(6\sqrt{21})+2(-8)}{2}$

Step:2.3

Take $2$ out as a common factor. $\frac{2(6\sqrt{21}-8)}{2}$

Step:2.4

Eliminate the common factors.

Step:2.4.1

Factor out the $2$ in the denominator. $\frac{2(6\sqrt{21}-8)}{2(1)}$

Step:2.4.2

Reduce the common factor. $\frac{\cancel{2}(6\sqrt{21}-8)}{\cancel{2}\cdot 1}$

Step:2.4.3

Simplify the expression. $\frac{6\sqrt{21}-8}{1}$

Step:2.4.4

Divide the numerator by $1$. $6\sqrt{21}-8$

Step:3

Present the final result in various formats.

Exact Form: $6\sqrt{21}-8$ Decimal Form: $19.49545416\dots$

Knowledge Notes:

To simplify an expression involving square roots and fractions, we follow these steps:

1. Simplify the Denominator: If the denominator is a square root of a perfect square, it can be simplified by taking the square root of that number.

2. Factor and Cancel: If there is a common factor in the numerator and the denominator, it can be factored out and then canceled to simplify the expression.

3. Simplify the Numerator: The numerator can often be simplified by factoring out common terms or simplifying the terms within the square root.

4. Final Simplification: Once the common factors are canceled, the expression can be simplified further, if possible, to get the final result.

5. Exact vs. Decimal Form: The final answer can be presented in exact form (with square roots and integers) or in decimal form, which requires the use of a calculator to approximate the square roots.

In this problem, we used the fact that $\sqrt{2^2}=2$ to simplify the denominator and then factored out a common factor of $2$ from the numerator and denominator to simplify the expression. The final result is presented in both exact and decimal forms.