The discussion focuses on the calculation of the sixth root of 3, specifically addressing the incorrect interpretation of the expression involving cube roots. The correct approach involves using the property of exponents where the sum of exponents applies when bases are the same. The participants clarify that the sixth root of 3 can be expressed as 3^(1/6), and the calculation of cube roots leads to the conclusion that 3^(1/3) * 3^(1/3) equals 3^(2/3), not 3^(1/6). This highlights the importance of correctly applying exponent rules in mathematical expressions.
PREREQUISITESStudents, educators, and anyone interested in improving their understanding of algebraic concepts, particularly in the manipulation of roots and exponents.
MarkFL said:In general, you want to use the following rule:
$$\sqrt[3]{a}\cdot\sqrt[3]{b}=\sqrt[3]{ab}$$
Using exponents, we can get the same result:
$$a^c\cdot b^c=(ab)^c$$
Now, when the base is the same, we can simply add exponents:
$$a^b\cdot a^c=a^{b+c}$$
So, in the given expression, we may write:
$$\sqrt[3]{3}\cdot\sqrt[3]{3}=\sqrt[3]{3\cdot3}=\sqrt[3]{3^2}=3^{\frac{2}{3}}=3^{\frac{1}{3}+\frac{1}{3}}=3^{\frac{1}{3}}\cdot3^{\frac{1}{3}}=\sqrt[3]{3}\cdot\sqrt[3]{3}$$
RTCNTC said:Great but is my answer wrong?
MarkFL said:Yes, your result is incorrect. :D
You want to add the two exponents to get 1/3 + 1/3 = 2/3.