what is the cube root of -729a9b6
The Correct Answer and Explanation is:
Correct Answer:
The cube root of −729a9b6-729a^9b^6 is: −9a3b2\boxed{-9a^3b^2}
To find the cube root of an expression like −729a9b6-729a^9b^6, we apply the cube root to each factor separately, using the property: xyz3=x3⋅y3⋅z3\sqrt[3]{xyz} = \sqrt[3]{x} \cdot \sqrt[3]{y} \cdot \sqrt[3]{z}
Step 1: Cube root of the constant
−729-729 is a negative number. The cube root of a negative number is negative (since (−a)3=−a3(-a)^3 = -a^3). Now, find the cube root of 729: 729=93so7293=9729 = 9^3 \quad \text{so} \quad \sqrt[3]{729} = 9
Thus, −7293=−9\sqrt[3]{-729} = -9
Step 2: Cube root of a9a^9
For exponents, we use the rule: a93=a9/3=a3\sqrt[3]{a^9} = a^{9/3} = a^3
Step 3: Cube root of b6b^6
Similarly, b63=b6/3=b2\sqrt[3]{b^6} = b^{6/3} = b^2
Step 4: Combine the results
Now combine all cube roots: −729a9b63=−7293⋅a93⋅b63=−9⋅a3⋅b2=−9a3b2\sqrt[3]{-729a^9b^6} = \sqrt[3]{-729} \cdot \sqrt[3]{a^9} \cdot \sqrt[3]{b^6} = -9 \cdot a^3 \cdot b^2 = \boxed{-9a^3b^2}
Summary:
This simplification uses basic properties of cube roots and exponents:
- xn3=xn/3\sqrt[3]{x^n} = x^{n/3}
- The cube root of a negative number is negative.
- Cube roots distribute over multiplication.
So the cube root of the entire expression −729a9b6-729a^9b^6 is −9a3b2\boxed{-9a^3b^2}.
