Simplify (12 square root of 21-16)/( square root of 4)
The problem presents an expression that consists of a numerator (12 multiplied by the square root of 21, then subtracting 16) and a denominator (the square root of 4). The question asks to simplify this mathematical expression by performing the indicated operations and reducing the expression to its simplest form. Simplification here likely involves carrying out the subtraction in the numerator, simplifying the square root in the denominator, and then simplifying the resulting fraction if possible.
$\frac{12 \sqrt{21} - 16}{\sqrt{4}}$
Answer
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: $19.49545416 \ldots$
Knowledge Notes:
To simplify an expression involving square roots and fractions, we follow these steps:
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.
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.
Simplify the Numerator: The numerator can often be simplified by factoring out common terms or simplifying the terms within the square root.
Final Simplification: Once the common factors are canceled, the expression can be simplified further, if possible, to get the final result.
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.
