Math

Question16. Approximate 2\sqrt{2} as 1.414. Find approximate values for: a) 200\sqrt{200} b) 20000\sqrt{20000}
17. Approximate 23\sqrt[3]{2} as 1.260 and 43\sqrt[3]{4} as 1.587. Find approximate values for: a) 20003\sqrt[3]{2000} b) 40000003\sqrt[3]{4000000}

Studdy Solution

STEP 1

Assumptions
1. An approximate value for 2\sqrt{2} is given as 1.414213562.
2. An approximate value for 23\sqrt[3]{2} is given as 1.25992105.
3. An approximate value for 43\sqrt[3]{4} is given as 1.587401052.
4. We are to find the approximate values for 200\sqrt{200}, 20000\sqrt{20000}, 20003\sqrt[3]{2000}, and 40000003\sqrt[3]{4000000} without using a calculator.
5. We will use properties of radicals to simplify the expressions.

STEP 2

To find 200\sqrt{200}, we can express 200 as a product of 2 and 100, which is a perfect square. We can then use the property of radicals that ab=ab\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}.
200=2100\sqrt{200} = \sqrt{2 \cdot 100}

STEP 3

We know that 100=10\sqrt{100} = 10 because 100 is a perfect square. So we can write:
200=2100\sqrt{200} = \sqrt{2} \cdot \sqrt{100}

STEP 4

Substitute the approximate value for 2\sqrt{2} and the exact value for 100\sqrt{100}.
2001.41421356210\sqrt{200} \approx 1.414213562 \cdot 10

STEP 5

Calculate the approximate value for 200\sqrt{200}.
2001.41421356210=14.14213562\sqrt{200} \approx 1.414213562 \cdot 10 = 14.14213562

STEP 6

To find 20000\sqrt{20000}, we can express 20000 as a product of 2 and 10000, which is a perfect square. We can then use the property of radicals that ab=ab\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}.
20000=210000\sqrt{20000} = \sqrt{2 \cdot 10000}

STEP 7

We know that 10000=100\sqrt{10000} = 100 because 10000 is a perfect square. So we can write:
20000=210000\sqrt{20000} = \sqrt{2} \cdot \sqrt{10000}

STEP 8

Substitute the approximate value for 2\sqrt{2} and the exact value for 10000\sqrt{10000}.
200001.414213562100\sqrt{20000} \approx 1.414213562 \cdot 100

STEP 9

Calculate the approximate value for 20000\sqrt{20000}.
200001.414213562100=141.4213562\sqrt{20000} \approx 1.414213562 \cdot 100 = 141.4213562

STEP 10

To find 20003\sqrt[3]{2000}, we can express 2000 as a product of 2 and 1000. We can then use the property of radicals that ab3=a3b3\sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b}.
20003=210003\sqrt[3]{2000} = \sqrt[3]{2 \cdot 1000}

STEP 11

We know that 10003=10\sqrt[3]{1000} = 10 because 1000 is a perfect cube. So we can write:
20003=2310003\sqrt[3]{2000} = \sqrt[3]{2} \cdot \sqrt[3]{1000}

STEP 12

Substitute the approximate value for 23\sqrt[3]{2} and the exact value for 10003\sqrt[3]{1000}.
200031.2599210510\sqrt[3]{2000} \approx 1.25992105 \cdot 10

STEP 13

Calculate the approximate value for 20003\sqrt[3]{2000}.
200031.2599210510=12.5992105\sqrt[3]{2000} \approx 1.25992105 \cdot 10 = 12.5992105

STEP 14

To find 40000003\sqrt[3]{4000000}, we can express 4000000 as a product of 4 and 1000000. We can then use the property of radicals that ab3=a3b3\sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b}.
40000003=410000003\sqrt[3]{4000000} = \sqrt[3]{4 \cdot 1000000}

STEP 15

We know that 10000003=100\sqrt[3]{1000000} = 100 because 1000000 is a perfect cube. So we can write:
40000003=4310000003\sqrt[3]{4000000} = \sqrt[3]{4} \cdot \sqrt[3]{1000000}

STEP 16

Substitute the approximate value for 43\sqrt[3]{4} and the exact value for 10000003\sqrt[3]{1000000}.
400000031.587401052100\sqrt[3]{4000000} \approx 1.587401052 \cdot 100

STEP 17

Calculate the approximate value for 40000003\sqrt[3]{4000000}.
400000031.587401052100=158.7401052\sqrt[3]{4000000} \approx 1.587401052 \cdot 100 = 158.7401052
The approximate values for each radical are as follows: a) 20014.14213562\sqrt{200} \approx 14.14213562 b) 20000141.4213562\sqrt{20000} \approx 141.4213562 a) 2000312.5992105\sqrt[3]{2000} \approx 12.5992105 b) 40000003158.7401052\sqrt[3]{4000000} \approx 158.7401052

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