Simplify each cube root expression. Describe the simplified form of the expression as rational or irrational. In your final answer, include all of your work. 3 square root of 81 3 square root of -64
The Correct Answer and Explanation is:
Problem 1: Simplify the cube root of 81.
The expression we are asked to simplify is 813\sqrt[3]{81}381.
Step 1: Express 81 in terms of its prime factors.
We start by factoring 81: 81=3481 = 3^481=34
Step 2: Apply the cube root.
The cube root of a number xxx, denoted x3\sqrt[3]{x}3x, is the number that, when multiplied by itself three times, equals xxx. Now, we apply the cube root to 81=3481 = 3^481=34: 813=343\sqrt[3]{81} = \sqrt[3]{3^4}381=334
We can rewrite this as: 343=343\sqrt[3]{3^4} = 3^{\frac{4}{3}}334=334
Step 3: Simplify the exponent.
To simplify 3433^{\frac{4}{3}}334, we separate the exponent into two parts: 343=31+13=31×3133^{\frac{4}{3}} = 3^{1 + \frac{1}{3}} = 3^1 \times 3^{\frac{1}{3}}334=31+31=31×331 31=3and33=3133^1 = 3 \quad \text{and} \quad \sqrt[3]{3} = 3^{\frac{1}{3}}31=3and33=331
Therefore, the simplified expression is: 813=3×33\sqrt[3]{81} = 3 \times \sqrt[3]{3}381=3×33
Conclusion for Problem 1:
The cube root of 81 simplifies to 3×333 \times \sqrt[3]{3}3×33. Since 33\sqrt[3]{3}33 is irrational, the entire expression is irrational. Therefore, the final answer is 3×333 \times \sqrt[3]{3}3×33.
Problem 2: Simplify the cube root of -64.
We are given the expression −643\sqrt[3]{-64}3−64.
Step 1: Express -64 as a product of prime factors.
The prime factorization of 64 is: 64=2664 = 2^664=26
Since we have a negative sign, we can write: −64=−1×64=−1×26-64 = -1 \times 64 = -1 \times 2^6−64=−1×64=−1×26
Step 2: Apply the cube root.
Now, we find the cube root of −64-64−64: −643=−1×263\sqrt[3]{-64} = \sqrt[3]{-1 \times 2^6}3−64=3−1×26
We can separate this into two parts: −13×263\sqrt[3]{-1} \times \sqrt[3]{2^6}3−1×326
Step 3: Simplify each part.
The cube root of -1 is −1-1−1, and the cube root of 262^626 is: 263=22=4\sqrt[3]{2^6} = 2^2 = 4326=22=4
Thus, the simplified expression is: −643=−1×4=−4\sqrt[3]{-64} = -1 \times 4 = -43−64=−1×4=−4
Conclusion for Problem 2:
The cube root of -64 simplifies to −4-4−4, which is a rational number.
Final Answer:
- 813=3×33\sqrt[3]{81} = 3 \times \sqrt[3]{3}381=3×33, which is irrational.
- −643=−4\sqrt[3]{-64} = -43−64=−4, which is rational.
