Simplify.
216^1/3
The Correct Answer and Explanation is:
To simplify 2161/3216^{1/3}2161/3, we are asked to find the cube root of 216. This means we need to determine which number, when multiplied by itself three times (cubed), gives 216.
Step 1: Understanding Cube Roots
The cube root of a number xxx, denoted x1/3x^{1/3}x1/3, is a value that, when raised to the third power (multiplied by itself three times), gives xxx. Mathematically, this can be expressed as:x3=ysuch thaty3=x\sqrt[3]{x} = y \quad \text{such that} \quad y^3 = x3x=ysuch thaty3=x
For example, 83=2\sqrt[3]{8} = 238=2 because 23=82^3 = 823=8.
Step 2: Prime Factorization of 216
To simplify 2161/3216^{1/3}2161/3, we begin by finding the prime factorization of 216. We divide 216 by the smallest prime numbers until we reach a product of primes:
- 216÷2=108216 \div 2 = 108216÷2=108 (since 216 is even, we start with 2).
- 108÷2=54108 \div 2 = 54108÷2=54.
- 54÷2=2754 \div 2 = 2754÷2=27.
- 27÷3=927 \div 3 = 927÷3=9 (since 27 is divisible by 3).
- 9÷3=39 \div 3 = 39÷3=3.
- 3÷3=13 \div 3 = 13÷3=1.
Thus, the prime factorization of 216 is:216=23×33216 = 2^3 \times 3^3216=23×33
Step 3: Applying the Cube Root
Now, we take the cube root of both sides of the equation 216=23×33216 = 2^3 \times 3^3216=23×33:2163=23×333\sqrt[3]{216} = \sqrt[3]{2^3 \times 3^3}3216=323×33
Since a33=a\sqrt[3]{a^3} = a3a3=a, we can simplify this as:2163=2×3=6\sqrt[3]{216} = 2 \times 3 = 63216=2×3=6
Final Answer:
Thus, 2161/3=6216^{1/3} = 62161/3=6.
Explanation of Cube Roots:
Taking the cube root is a way to reverse cubing a number. Just as the square root is the reverse of squaring a number, the cube root is the reverse of cubing. This is why simplifying cube roots often involves factoring the number into powers of prime numbers. The cube root of a number is particularly useful in geometry and algebra when dealing with volumes of cubes and cubic equations.