find the rationalizing factor of cube root of 81 ……. process please
The Correct Answer and Explanation is:
To find the rationalizing factor of the cube root of 81, we need to first understand what it means to “rationalize” the cube root. Rationalizing a radical expression means transforming it into a form where the denominator no longer has a radical (in this case, cube roots) present.
Step-by-Step Process
- Expression Setup:
We are working with the cube root of 81, which is written as:
813\sqrt[3]{81}381 - Factorization of 81:
Start by factoring 81 into its prime factors.
81=3481 = 3^481=34
Thus, we can rewrite the cube root of 81 as:
813=343\sqrt[3]{81} = \sqrt[3]{3^4}381=334 - Simplifying the Cube Root:
We know that the cube root of ana^nan is an/3a^{n/3}an/3. Applying this property to 343^434, we get:
343=34/3\sqrt[3]{3^4} = 3^{4/3}334=34/3 Now, we can break this down as:
34/3=31+1/3=3×31/33^{4/3} = 3^{1 + 1/3} = 3 \times 3^{1/3}34/3=31+1/3=3×31/3 So, the cube root of 81 simplifies to:
813=3×33\sqrt[3]{81} = 3 \times \sqrt[3]{3}381=3×33 - Rationalizing the Cube Root:
To rationalize the expression, we want to remove the cube root from the denominator if it were in a fraction. For example, if we had 1813\frac{1}{\sqrt[3]{81}}3811, we would multiply both the numerator and denominator by 323=93\sqrt[3]{3^2} = \sqrt[3]{9}332=39, since multiplying the cube roots gives:
33×93=273=3\sqrt[3]{3} \times \sqrt[3]{9} = \sqrt[3]{27} = 333×39=327=3 Thus, the rationalizing factor for 813\sqrt[3]{81}381 (or for 343\sqrt[3]{3^4}334) is 93\sqrt[3]{9}39.
Explanation of Rationalizing Process
Rationalizing the cube root essentially involves multiplying by an expression that will eliminate the radical in the denominator. In the case of cube roots, the goal is to make the exponent of the base inside the cube root a multiple of 3. This process often requires multiplying by a number that compensates for the fraction in the exponent.
In the example above, we saw that to eliminate the cube root in the denominator, we multiply by 93\sqrt[3]{9}39, which has the effect of transforming the cube root into a whole number. This process works similarly for square roots and higher-order roots, but with the respective powers required to eliminate the radical.
So, the rationalizing factor of the cube root of 81 is 93\sqrt[3]{9}39, which allows the expression to be simplified into a more manageable form without a cube root in the denominator.
