Question

what is the following product? assume ( d geq 0 ).
( sqrt3{d} cdot sqrt3{d} cdot sqrt3{d} )
options: ( d^3 ), ( d ), ( sqrt3{d} ), ( 3left( sqrt3{d}
ight) )

Explanation:

Step1: Recall the property of radicals

The cube root of a number \( \sqrt[3]{x} \) can be written as \( x^{\frac{1}{3}} \). So, \( \sqrt[3]{d}=d^{\frac{1}{3}} \).

Step2: Multiply the three cube roots

We have \( \sqrt[3]{d}\cdot\sqrt[3]{d}\cdot\sqrt[3]{d} \). Using the property of exponents \( a^m\cdot a^n = a^{m + n} \), here \( m=n=\frac{1}{3} \) for each term. So, adding the exponents: \( \frac{1}{3}+\frac{1}{3}+\frac{1}{3}=\frac{3}{3} = 1\)? Wait, no, wait. Wait, actually, \( \sqrt[3]{d}\cdot\sqrt[3]{d}\cdot\sqrt[3]{d}=(d^{\frac{1}{3}})\cdot(d^{\frac{1}{3}})\cdot(d^{\frac{1}{3}}) \). Using the exponent rule \( (a^m)^n=a^{mn} \) when multiplying same bases, it's \( d^{\frac{1}{3}+\frac{1}{3}+\frac{1}{3}}=d^{\frac{3}{3}} = d^1=d \)? Wait, no, that's not right. Wait, no, the cube root of \( d \) multiplied three times: \( \sqrt[3]{d}\times\sqrt[3]{d}\times\sqrt[3]{d}=\sqrt[3]{d\times d\times d}=\sqrt[3]{d^3} \). And \( \sqrt[3]{d^3}=d \), because the cube root and cubing are inverse operations. Wait, but also, another way: \( (\sqrt[3]{d})^3 \). Because multiplying three of the same cube roots is the same as cubing the cube root. And \( (\sqrt[3]{d})^3=d \), since \( (\sqrt[3]{x})^3=x \) for any \( x \). Wait, but let's check the options. Wait, the options are \( d^3 \), \( d \), \( 3(\sqrt[3]{d}) \), and \( \sqrt[3]{3d} \). Wait, no, wait, maybe I made a mistake. Wait, no, \( \sqrt[3]{d}\cdot\sqrt[3]{d}\cdot\sqrt[3]{d}=(d^{\frac{1}{3}})^3 \) (since we have three factors of \( d^{\frac{1}{3}} \), which is \( (d^{\frac{1}{3}})^3 \)). Using the exponent rule \( (a^m)^n=a^{mn} \), so \( (d^{\frac{1}{3}})^3=d^{\frac{1}{3}\times3}=d^1 = d \). Wait, but also, \( \sqrt[3]{d}\times\sqrt[3]{d}\times\sqrt[3]{d}=\sqrt[3]{d\times d\times d}=\sqrt[3]{d^3}=d \). So the product is \( d \). Wait, but let's check the options. The options are: top left \( d^3 \), top right \( d \), bottom left \( 3(\sqrt[3]{d}) \), bottom right \( \sqrt[3]{3d} \). Wait, no, the image shows the options as: first (top right) \( d^3 \), second (middle right) \( d \), third (top left) \( 3(\sqrt[3]{d}) \), fourth (bottom right) \( \sqrt[3]{3d} \)? Wait, maybe I misread. Wait, the problem is \( \sqrt[3]{d}\cdot\sqrt[3]{d}\cdot\sqrt[3]{d} \). Let's compute again. \( \sqrt[3]{d}=d^{1/3} \), so multiplying three of them: \( d^{1/3}\times d^{1/3}\times d^{1/3}=d^{(1/3 + 1/3 + 1/3)}=d^{1}=d \). So the product is \( d \).

Answer:

\( d \) (the middle right option)