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Chapter 2: Problem 147

Suppose you have a calculator that can only compute square roots and can multiply. Explain how you could use this calculator to compute \(7^{3 / 4}\).

Short Answer

Expert verified
To compute \(7^{3 / 4}\) with a calculator that can only compute square roots and multiplication, rewrite the expression as \((7^{1 / 4})^3\). This involves finding the fourth root of 7, which can be achieved by finding the square root of the square root, i.e., \(\sqrt{\sqrt{7}}\). Use the calculator's square root function twice to compute this result, then cube the result, i.e., multiply it by itself three times, yielding the final value of \(7^{3 / 4}\).

Step by step solution

01

Rewrite the exponent as (7^(1/4))^3

Rewrite the exponent \(7^{3 / 4}\) as \((7^{1 / 4})^3\). This will allow us to find the fourth root of 7 using the calculator's square root function and then multiply the result by itself three times.
02

Find the square root of 7

Using the calculator's square root function, find \(\sqrt{7}\).
03

Find the square root of the result from Step 2

To find the fourth root of 7, we need to find the square root of the result from step 2, which is \(\sqrt{\sqrt{7}}\). Use the calculator's square root function again on the result from step 2.
04

Cube the result from Step 3

Now that we have found \(7^{1 / 4}\) or the fourth root of 7 from step 3, we can cube this result, i.e., multiply the result from step 3 by itself three times. Let the result from step 3 be \(x\), then multiply \(x\times x \times x\).
05

Final result

The result from step 4 is the value of \(7^{3 / 4}\).

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Most popular questions from this chapter

Give an example of polynomials \(p\) and \(q\) of degree 3 such that \(p(1)=q(1), p(2)=q(2),\) and \(p(3)=q(3),\) but \(p(4) \neq q(4)\).

Verify that \((x+y)^{3}=x^{3}+3 x^{2} y+3 x y^{2}+y^{3}\).

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r s)(x) $$

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r-s)(x) $$

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (t \circ r)(x) $$
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