Question

4 from unit 4, lesson 3
suppose that a friend missed class and never learned what ( 37^{\frac{1}{3}} ) means.
a. use exponent rules that your friend would already know to show your friend how to calculate ( left(37^{\frac{1}{3}}
ight)^{3} ).
b. explain why this means that ( 37^{\frac{1}{3}} ) is the cube root of 37.

Explanation:

Step1: Use the power - of - a - power rule

The power - of - a - power rule states that \((a^{m})^{n}=a^{mn}\). For \((37^{\frac{1}{3}})^{3}\), where \(a = 37\), \(m=\frac{1}{3}\), and \(n = 3\), we have \((37^{\frac{1}{3}})^{3}=37^{\frac{1}{3}\times3}\).

Step2: Simplify the exponent

Calculate \(\frac{1}{3}\times3\). Since \(\frac{1}{3}\times3 = 1\), then \(37^{\frac{1}{3}\times3}=37^{1}\).

Step3: Evaluate the result

By the definition of \(a^{1}=a\) (where \(a\) is a real number), when \(a = 37\), \(37^{1}=37\).

Part b

Let \(x = 37^{\frac{1}{3}}\). If we raise \(x\) to the power of \(3\), \((x)^{3}=(37^{\frac{1}{3}})^{3}\). From part (a), \((37^{\frac{1}{3}})^{3}=37\). By the definition of the cube root, if \(y^{3}=z\), then \(y=\sqrt[3]{z}\). Here, since \(x^{3}=37\), then \(x = 37^{\frac{1}{3}}=\sqrt[3]{37}\)

Answer:

a. \((37^{\frac{1}{3}})^{3}=37^{\frac{1}{3}\times3}=37^{1}=37\)
b. Let \(x = 37^{\frac{1}{3}}\). Since \((x)^{3}=37\), by the definition of the cube root (\(y^{3}=z\) implies \(y = \sqrt[3]{z}\)), \(37^{\frac{1}{3}}\) is the cube root of \(37\), i.e., \(37^{\frac{1}{3}}=\sqrt[3]{37}\)