Problem 7 - Analysis 1: Ein Lehr- und Arbeitsbuch für Studienanfänger

Question

Welche der folgenden Gleichungen sind richtig, welche falsch? a) \(^{3} \sqrt{27}=3\) b) \(^{3} \sqrt{-27}=-3\) c) \(-\sqrt[3]{-27}=3\), d) \(^{3} \sqrt{-27}=3\) c) \(^{6} \sqrt{-64}=-2\), f) \(^{6} \sqrt{64}=2\), g) \({ }^{6} \sqrt{(-2)^{6}}=2\), h) \(^{4} \sqrt{1,4641}=1,1\), i) \(^{4} \sqrt{(a-b)^{4}}=b-a .\)

Step-by-Step Solution

Answer

Correct: a, b, f, g, h. Incorrect: c, d, e, i.

Step 1: Evaluate Equation a

Let's evaluate the cube root of 27. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. Here, \[^{3} \sqrt{27} = 3\] because \[3 \times 3 \times 3 = 27.\] This equation is correct.

Step 2: Evaluate Equation b

Evaluate the cube root of -27. The cube root of a negative number is negative, so:\[^{3} \sqrt{-27} = -3\]because\[-3 \times -3 \times -3 = -27.\]This equation is correct.

Step 3: Evaluate Equation c

Here, we're looking at:\[-\sqrt[3]{-27} = 3.\]Calculate the cube root of -27:\[\sqrt[3]{-27} = -3,\]so:\[-\sqrt[3]{-27} = -(-3) = 3,\] making this equation incorrect.

Step 4: Evaluate Equation d

Evaluate the cube root of -27:\[^{3} \sqrt{-27} = 3.\]As established:\[^{3} \sqrt{-27} = -3,\]so this equation is incorrect.

Step 5: Evaluate Equation e

For:\[^{6} \sqrt{-64} = -2.\]The sixth root of -64 is not defined in the real numbers because an even root of a negative number is not defined. So, this equation is incorrect.

Step 6: Evaluate Equation f

Evaluate the sixth root of 64. The sixth root of a number is a value that, when raised to the power of six, equals the original number:\[^{6} \sqrt{64} = 2,\]because:\[2^6 = 64.\]Thus, this equation is correct.

Step 7: Evaluate Equation g

Calculate:\[^{6} \sqrt{(-2)^{6}} = 2.\]Since \[(-2)^6 = 64,\]we have \[^{6} \sqrt{64} = 2.\]This equation is correct.

Step 8: Evaluate Equation h

Check:\[^{4} \sqrt{1.4641} = 1.1.\]Compute the fourth root of 1.4641:\[1.1^4 = 1.4641,\]so this equation is correct.

Step 9: Evaluate Equation i

Evaluate:\[^{4} \sqrt{(a-b)^{4}} = b-a.\]Since:\[(a-b)^4,\]taking the fourth root gives:\[|a-b|,\]this is not generally equal to \(b-a\) unless \(a\le b\).Therefore, this equation is incorrect.

Key Concepts

Cube Roots

Cube roots are quite intriguing. When we talk about the cube root, we are searching for a number, which when multiplied by itself three times, will return the original value. For instance, the cube root of 27 is 3 because when you calculate \(3 \times 3 \times 3\), you get 27. It is important to note that cube roots accept both positive and negative numbers since multiplying a negative number three times results in a negative product as well. Therefore, the cube root of negative numbers produces negative results, such as \( \sqrt[3]{-27} = -3\), because \(-3 \times -3 \times -3 = -27\). Something interesting to remember: a cube root of 0 is always 0. This simplicity makes understanding cube roots less daunting.

Sixth Roots

The concept of sixth roots can be a bit puzzling. Simply put, the sixth root of a number is a value that, when raised to the sixth power, results in the original number. Consider the number 64. The sixth root of 64 is 2 because \(2^6 = 64\). Generally, sixth roots of positive numbers are quite straightforward.
However, it's different for negative numbers. For negative values, sixth roots are not defined in the real numbers. This is due to the fact that any even root of a negative number results in a contradiction in real number operations. So, when faced with a problem like \(^{6}\sqrt{-64}\), remember it doesn't have a solution within the real number system.

Fourth Roots

Fourth roots involve a sequence where a number, when raised to the power of four, regains the original number. If you take a real number such as 1.4641, its fourth root would be 1.1, as \(1.1^4 = 1.4641\).Moreover, for any expression such as \((a-b)^4\), you take the fourth root which results in \(|a-b|\). This is crucial to understand because it reflects the absolute value. Therefore, when calculating with variables, make sure you focus on absolute values; it helps prevent incorrect conclusions like interpreting \(\sqrt[4]{(a-b)^4} = b - a\), which is usually not a valid equation unless specific conditions hold (such as \(a \le b\)).
This careful approach helps to handle more complex algebraic expressions efficiently.

Real Numbers

Real numbers are the tapestry of mathematics, covering all the rational and irrational numbers that fit seamlessly on the number line. This comprehensive set includes quantities we use daily — integers like 0, 1, -1, fractions like 1/2, and irrational numbers like \(\pi\).
In the context of roots, understanding real numbers helps identify which operations are feasible. For instance, finding the square root of a negative number such as \(-1\) is not possible within the real number system, leading to the introduction of imaginary numbers. Similarly, an even root of any negative value has no real solution. Hence, knowing the scope of real numbers aids in avoiding errors related to undefined operations. Their versatility across mathematics makes them essential in grasping the full spectrum of number operations and equations, from simple root evaluations to complex algebraic functions.