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Math  /  Numbers & Operations

Question48162,8448,56112,−4284,36218,−2040,24288,−56126\frac{48}{\sqrt{162}}, \quad \frac{84}{\sqrt{48}}, \quad \frac{56}{\sqrt{112}}, \quad \frac{-42}{\sqrt{84}}, \quad \frac{36}{2 \sqrt{18}}, \quad \frac{-20}{\sqrt{40}}, \quad \frac{24}{\sqrt{288}}, \quad \frac{-56}{\sqrt{126}}

Studdy Solution

STEP 1

1. Each expression is a fraction with a numerator and a denominator involving square roots.
2. The task is to simplify each expression.
3. Simplification involves rationalizing the denominator and reducing the fraction if possible.

STEP 2

1. Simplify the square root in the denominator.
2. Rationalize the denominator.
3. Simplify the fraction.

STEP 3

Simplify the square root in the denominator for each expression.
1. 162=81×2=92 \sqrt{162} = \sqrt{81 \times 2} = 9\sqrt{2}
2. 48=16×3=43 \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}
3. 112=16×7=47 \sqrt{112} = \sqrt{16 \times 7} = 4\sqrt{7}
4. 84=4×21=221 \sqrt{84} = \sqrt{4 \times 21} = 2\sqrt{21}
5. 18=9×2=32 \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}
6. 40=4×10=210 \sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10}
7. 288=144×2=122 \sqrt{288} = \sqrt{144 \times 2} = 12\sqrt{2}
8. 126=9×14=314 \sqrt{126} = \sqrt{9 \times 14} = 3\sqrt{14}

STEP 4

Rationalize the denominator for each expression.
1. 4892×22=48218 \frac{48}{9\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{48\sqrt{2}}{18}
2. 8443×33=84312 \frac{84}{4\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{84\sqrt{3}}{12}
3. 5647×77=56728 \frac{56}{4\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{56\sqrt{7}}{28}
4. −42221×2121=−422142 \frac{-42}{2\sqrt{21}} \times \frac{\sqrt{21}}{\sqrt{21}} = \frac{-42\sqrt{21}}{42}
5. 362⋅32×22=36212 \frac{36}{2 \cdot 3\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{36\sqrt{2}}{12}
6. −20210×1010=−201020 \frac{-20}{2\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \frac{-20\sqrt{10}}{20}
7. 24122×22=24224 \frac{24}{12\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{24\sqrt{2}}{24}
8. −56314×1414=−561442 \frac{-56}{3\sqrt{14}} \times \frac{\sqrt{14}}{\sqrt{14}} = \frac{-56\sqrt{14}}{42}

STEP 5

Simplify the fraction for each expression.
1. 48218=823 \frac{48\sqrt{2}}{18} = \frac{8\sqrt{2}}{3}
2. 84312=73 \frac{84\sqrt{3}}{12} = 7\sqrt{3}
3. 56728=27 \frac{56\sqrt{7}}{28} = 2\sqrt{7}
4. −422142=−21 \frac{-42\sqrt{21}}{42} = -\sqrt{21}
5. 36212=32 \frac{36\sqrt{2}}{12} = 3\sqrt{2}
6. −201020=−10 \frac{-20\sqrt{10}}{20} = -\sqrt{10}
7. 24224=2 \frac{24\sqrt{2}}{24} = \sqrt{2}
8. −561442=−4143 \frac{-56\sqrt{14}}{42} = -\frac{4\sqrt{14}}{3}
The simplified expressions are:
1. 823 \frac{8\sqrt{2}}{3}
2. 73 7\sqrt{3}
3. 27 2\sqrt{7}
4. −21 -\sqrt{21}
5. 32 3\sqrt{2}
6. −10 -\sqrt{10}
7. 2 \sqrt{2}
8. −4143 -\frac{4\sqrt{14}}{3}

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