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Chapter 3: Problem 86

Suppose you have a calculator that can only compute square roots. Explain how you could use this calculator to compute \(7^{1 / 8}\).

Short Answer

Expert verified
To compute \(7^{1/8}\) using a calculator with only square root functionality, express \(7^{1/8}\) as a series of square roots: \(((7^{1/8})^{1/2})^{1/2}\). Compute the square root of 7, then take the square root of the result twice. Doing this, we get \(7^{1/8} \approx 1.276055\).

Step by step solution

01

Rewrite the expression as repeated square roots

Express \(7^{1/8}\) as a series of square roots. Notice that since \(8 = 2^3\), the 8th root of 7 can be written as: \[ 7^{1/8} = (7^{1/2})^{1/4} = ((7^{1/8})^{1/2})^{1/2} \]
02

Compute the square root of 7

First, use the calculator to find the square root of 7: \[ \sqrt{7} \approx 2.645751 \]
03

Compute the square root of the previous result

Next, find the square root of the result from Step 2: \[ \sqrt{2.645751} \approx 1.626576 \]
04

Compute the square root of the previous result again

Finally, find the square root of the result from Step 3: \[ \sqrt{1.626576} \approx 1.276055 \]
05

Conclusion

So, using the calculator that can only compute square roots, we find that \(7^{1/8} \approx 1.276055\).

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