First, let's simplify the expression inside the square root:
[ \sqrt{75 - 0.1 \sqrt{300 - 27} } ]
[ = \sqrt{75 - 0.1 \sqrt{273} } ]
Next, simplify the square root inside the square root:
[ = \sqrt{75 - 0.1 \times 3 \times \sqrt{91} } ]
[ = \sqrt{75 - 0.3 \sqrt{91} } ]
Now, the expression inside the square root is simplified. We can simplify it further:
[ = \sqrt{75} - \sqrt{0.3 \sqrt{91}} ]
[ = \sqrt{75} - \sqrt{0.3} \times \sqrt{\sqrt{91}} ]
[ = \sqrt{75} - \sqrt{0.3} \times \sqrt{91^{\frac{1}{2}}} ]
[ = 5\sqrt{3} - \sqrt{0.3} \times 9 ]
[ = 5\sqrt{3} - 3\sqrt{3} ]
[ = 2\sqrt{3} ]
Therefore, [ \sqrt{75 - 0.1 \sqrt{300 - 27} } = 2\sqrt{3} ]
First, let's simplify the expression inside the square root:
[ \sqrt{75 - 0.1 \sqrt{300 - 27} } ]
[ = \sqrt{75 - 0.1 \sqrt{273} } ]
Next, simplify the square root inside the square root:
[ = \sqrt{75 - 0.1 \times 3 \times \sqrt{91} } ]
[ = \sqrt{75 - 0.3 \sqrt{91} } ]
Now, the expression inside the square root is simplified. We can simplify it further:
[ = \sqrt{75} - \sqrt{0.3 \sqrt{91}} ]
[ = \sqrt{75} - \sqrt{0.3} \times \sqrt{\sqrt{91}} ]
[ = \sqrt{75} - \sqrt{0.3} \times \sqrt{91^{\frac{1}{2}}} ]
[ = 5\sqrt{3} - \sqrt{0.3} \times 9 ]
[ = 5\sqrt{3} - 3\sqrt{3} ]
[ = 2\sqrt{3} ]
Therefore, [ \sqrt{75 - 0.1 \sqrt{300 - 27} } = 2\sqrt{3} ]