√12 + 21/√3 can be written in the form m√3, where m is an integer. What is the value of m?
Understand the Problem
The question is asking to simplify the expression √12 + 21/√3 and rewrite it in the form m√3, where m is an integer. The goal is to find the value of m after performing the necessary simplifications.
Answer
The value of \( m \) is \( 9 \).
Answer for screen readers
The value of ( m ) is ( 9 ).
Steps to Solve
- Simplify the square root
Start by simplifying the first term, $\sqrt{12}$.
We know that ( \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} ).
- Simplify the fraction
Next, simplify the second term, ( \frac{21}{\sqrt{3}} ).
To eliminate the square root from the denominator, multiply the numerator and denominator by ( \sqrt{3} ):
$$ \frac{21}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{21\sqrt{3}}{3} = 7\sqrt{3}. $$
- Combine the terms
Now, combine the results from steps 1 and 2:
$$ 2\sqrt{3} + 7\sqrt{3} = (2 + 7)\sqrt{3} = 9\sqrt{3}. $$
Thus, we can write the expression as ( 9\sqrt{3} ), where ( m = 9 ).
The value of ( m ) is ( 9 ).
More Information
This problem involves simplifying square roots and fractions, which are common skills in algebra. The final expression shows that both terms can be combined to express the result in the desired format.
Tips
- Not simplifying square roots properly: Always check for perfect squares when simplifying square roots.
- Confusing division by a square root: Remember to multiply both the numerator and denominator by the square root to rationalize the expression.
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