Use the quotient property to simplify roots.
\sqrt{\frac{27a^{13}}{b^{12}}}
factor out all the perfect squares. Even powers are perfect squares.
√(27a^13/(b^12)
= √(9a^12 * 3a/b^12)
= 3a^6/b^6 √(3a)
To simplify the given expression using the quotient property of radicals, you can split the radical into two separate radicals by applying the property:
\sqrt{\frac{27a^{13}}{b^{12}}} = \frac{\sqrt{27a^{13}}}{\sqrt{b^{12}}}
Now, let's simplify each radical individually:
1. Simplifying the numerator:
- We can break down 27 into its prime factors as 3 * 3 * 3 = 3^3.
- Since we are taking the square root, we can take out one factor of 3 from the square root, leaving us with 3\sqrt{3}.
- For the variable, we can divide the exponent of a by 2 since we are taking the square root. So, we have a^{\frac{13}{2}}.
Therefore, the numerator simplifies to 3 \sqrt{3a^{\frac{13}{2}}}.
2. Simplifying the denominator:
- We can break down b^{12} into its prime factors as b * b * b * b * b * b * b * b * b * b * b * b = b^{6} * b^{6}.
- Since we are taking the square root, we can divide each exponent by 2, which gives us b^{6}.
Therefore, the denominator simplifies to \sqrt{b^{12}} = b^{6}.
Now, substituting these simplified forms back into the original expression, we have:
\frac{\sqrt{27a^{13}}}{\sqrt{b^{12}}} = \frac{3 \sqrt{3a^{\frac{13}{2}}}}{b^{6}}
Hence, the given expression is simplified using the quotient property of radicals as \frac{3 \sqrt{3a^{\frac{13}{2}}}}{b^{6}}.
To simplify the given expression using the quotient property of radicals, we can first simplify the numerator and denominator separately.
Starting with the numerator, we have:
√(27a^13)
The square root of 27 can be simplified as follows:
√(9 * 3) = 3√3
In the case of the variable exponent, we can write it as a product of square roots:
√(a^12 * a) = a^6√a
Now moving on to the denominator:
√(b^12)
Since the square root of a squared term is just the term itself, we can simplify this as:
b^6
Putting it all together, we have:
√(27a^13/b^12) = (3√3 * a^6√a) / b^6