Simplify the Radical Expression 3 square root of 12-5 square root of 27+2 square root of 75
The question provided is asking for the simplification of a mathematical expression that involves radical terms. Specifically, the expression contains three terms, each with coefficients in front of a square root. To simplify a radical expression like this one, you would typically look for ways to factor out perfect squares from under the radicals and combine like terms to reduce the expression to its simplest form. The terms involved are "3 square root of 12," "5 square root of 27," and "2 square root of 75," and the operations between them are subtraction and addition. The goal is to perform these operations resulting in an expression that contains radicals as simplified as possible or even possibly without any radicals if they can be eliminated through simplification.
$3 \sqrt{12} - 5 \sqrt{27} + 2 \sqrt{75}$
Answer
Solution:
Step:1
Break down each radical term into its simplest form.
Step:1.1
Express $12$ as a product of $2^{2} \cdot 3$.
Step:1.1.1
Extract the square factor from $12$: $3 \sqrt{4 \cdot 3} - 5 \sqrt{27} + 2 \sqrt{75}$
Step:1.1.2
Represent $4$ as $2^{2}$: $3 \sqrt{2^{2} \cdot 3} - 5 \sqrt{27} + 2 \sqrt{75}$
Step:1.2
Remove the square terms from under the radical: $3(2 \sqrt{3}) - 5 \sqrt{27} + 2 \sqrt{75}$
Step:1.3
Recognize that the absolute value of $2$ is $2$: $3(2 \sqrt{3}) - 5 \sqrt{27} + 2 \sqrt{75}$
Step:1.4
Combine the $2$ and $3$: $6 \sqrt{3} - 5 \sqrt{27} + 2 \sqrt{75}$
Step:1.5
Decompose $27$ into $3^{2} \cdot 3$.
Step:1.5.1
Isolate the square factor from $27$: $6 \sqrt{3} - 5 \sqrt{9 \cdot 3} + 2 \sqrt{75}$
Step:1.5.2
Express $9$ as $3^{2}$: $6 \sqrt{3} - 5 \sqrt{3^{2} \cdot 3} + 2 \sqrt{75}$
Step:1.6
Extract the square terms from under the radical: $6 \sqrt{3} - 5(3 \sqrt{3}) + 2 \sqrt{75}$
Step:1.7
The absolute value of $3$ is $3$: $6 \sqrt{3} - 5(3 \sqrt{3}) + 2 \sqrt{75}$
Step:1.8
Multiply $-5$ by $3$: $6 \sqrt{3} - 15 \sqrt{3} + 2 \sqrt{75}$
Step:1.9
Represent $75$ as $5^{2} \cdot 3$.
Step:1.9.1
Separate the square factor from $75$: $6 \sqrt{3} - 15 \sqrt{3} + 2 \sqrt{25 \cdot 3}$
Step:1.9.2
Rewrite $25$ as $5^{2}$: $6 \sqrt{3} - 15 \sqrt{3} + 2 \sqrt{5^{2} \cdot 3}$
Step:1.10
Pull out the square terms from under the radical: $6 \sqrt{3} - 15 \sqrt{3} + 2(5 \sqrt{3})$
Step:1.11
The absolute value of $5$ is $5$: $6 \sqrt{3} - 15 \sqrt{3} + 2(5 \sqrt{3})$
Step:1.12
Combine $2$ and $5$: $6 \sqrt{3} - 15 \sqrt{3} + 10 \sqrt{3}$
Step:2
Subtract $15 \sqrt{3}$ from $6 \sqrt{3}$: $-9 \sqrt{3} + 10 \sqrt{3}$
Step:3
Combine $-9 \sqrt{3}$ and $10 \sqrt{3}$: $\sqrt{3}$
Step:4
Present the result in various formats.
Exact Form: $\sqrt{3}$ Decimal Form: $1.73205080 \ldots$
Knowledge Notes:
To simplify a radical expression, one must follow these steps:
Factor the number under the radical into its prime factors or into perfect squares and other numbers if possible.
Use the property of radicals that $\sqrt{a^2} = |a|$ to pull out square factors from under the radical sign.
Simplify the expression by multiplying or dividing any coefficients outside the radicals.
Combine like terms, which are radical terms with the same radicand (the number under the radical sign).
The absolute value is used because the square root function only yields non-negative results, so when pulling out squares, we consider the absolute value of the number.
If the expression requires it, continue to simplify by adding or subtracting like terms.
The final expression can be presented in its exact radical form or as a decimal approximation if necessary.
