\( \sqrt[3]{7}=7^{\frac{1}{3}} \) A. \( \left(7^{\frac{1}{3}}\right)^{3}=7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}}=7 \cdot\left(\frac{1}{3}+\frac{1}{3}+\frac{1}{3}\right)=7 \cdot \frac{3}{3}=7 \cdot 1=7 \) B. \( \left(7^{\frac{1}{3}}\right)^{3}=7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}}=7 \cdot 7^{\frac{1}{3}}=3 \cdot \frac{1}{3} \cdot 7=1 \cdot 7=7 \) C. \( \left(7^{\frac{1}{3}}\right)^{3}=7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}}=7^{\frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3}}=7^{\frac{3}{3}}=7^{1}=7 \) D. \( \left(7^{\frac{1}{3}}\right)^{3}=7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}}=7^{\frac{1}{3}+\frac{1}{3}+\frac{1}{3}}=7^{\frac{3}{3}}=7^{1}=7 \)
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Let's travel back in time! Cube roots have been around for centuries, dating back to ancient civilizations, where mathematicians like the Babylonians devised methods to approximate values of cube roots. They laid the groundwork that would lead to our modern understanding of exponents and roots, using a blend of practical problem-solving and geometric intuitions. Imagine how mind-blowing it must have been for those early thinkers to realize that one number could be expressed in so many different, yet connected, ways! Now let’s apply our knowledge! Understanding cube roots can be incredibly useful in various fields, including computer graphics, physics, and even finance when dealing with exponential growth or decay. A practical approach is to remember \( a^{m/n} \) and use it to simplify calculations involving roots and exponents. By recognizing patterns in these expressions, you can quickly simplify complex problems and avoid common pitfalls like misapplying the properties of exponents. So next time you see a cube root, think of all the real-world applications it could relate to!
