Extracting Factors from Radicals
If the radicand includes a factor whose exponent is a multiple of the index, that factor can be extracted by dividing the exponent (m) by the index of the root (n). $$ \sqrt[n]{a^m \cdot a^q} = a^{\frac{m}{n}} \cdot \sqrt[n]{a^q} $$ This property is valid for a≥0. For even roots, the extracted factor must be non-negative.
If a factor has an exponent greater than the index (m>n) but is not a multiple of it, we can apply the laws of exponents to rewrite the power as a product and separate a term whose exponent is a multiple of the index.
$$ \sqrt[n]{a^m} = \sqrt[n]{a^{n \cdot q} \cdot a^r} = a^q \cdot \sqrt[n]{a^r} $$
Here, q is the quotient and r is the remainder when m is divided by n:
$$ m = n \cdot q + r $$
Note. If the sign of a factor is unknown, extracting it from an even root requires absolute value. $$ \sqrt[2]{2 \cdot a^2} = |a^{\frac{2}{2}}| \cdot \sqrt[2]{2} = |a| \cdot \sqrt{2} $$
A Practical Example
Consider the radical
$$ \sqrt[3]{a^6 \cdot b^2} $$
The factor a6 has an exponent greater than the index of the root (6>3). In addition, the exponent is a multiple of the index because 6 = 3·2.
Apply the product rule for radicals:
$$ \sqrt[3]{a^6} \cdot \sqrt[3]{b^2} $$
Now apply the invariant property of radicals.
Divide both the index and the exponent of the radicand in the first radical by 3:
$$ \sqrt[\frac{3}{3}]{a^{\frac{6}{3}}} \cdot \sqrt[3]{b^2} $$
$$ \sqrt[\frac{\not{3}}{\not{3}}]{a^2} \cdot \sqrt[3]{b^2} $$
This yields the extracted form:
$$ a^2 \cdot \sqrt[3]{b^2} $$
Note. The factor b cannot be extracted because its exponent (2) is smaller than the index of the root (3).
Example 2
Evaluate the same expression with a = 2 and b = 5:
$$ \sqrt[3]{2^6 \cdot 5^2} $$
Apply the product rule:
$$ \sqrt[3]{2^6} \cdot \sqrt[3]{5^2} $$
Use the invariant property on the first radical:
$$ \sqrt[\frac{3}{3}]{2^{\frac{6}{3}}} \cdot \sqrt[3]{5^2} $$
The first radical simplifies:
$$ 2^2 \cdot \sqrt[3]{5^2} $$
Final result:
$$ 4 \cdot \sqrt[3]{5^2} $$
Example 3
Consider the radical
$$ \sqrt[3]{a^7 \cdot b^2} $$
The exponent 7 exceeds the index 3 but is not a multiple of it.
Divide 7 by 3:
$$ 7 = 2 \cdot 3 + 1 $$
Rewrite the exponent:
$$ \sqrt[3]{a^{2 \cdot 3 + 1} \cdot b^2} $$
Apply the laws of exponents:
$$ \sqrt[3]{a^{2 \cdot 3} \cdot a \cdot b^2} $$
Apply the product rule for radicals:
$$ \sqrt[3]{a^{2 \cdot 3}} \cdot \sqrt[3]{a \cdot b^2} $$
Use the invariant property:
$$ \sqrt[\frac{3}{3}]{a^{\frac{2 \cdot 3}{3}}} \cdot \sqrt[3]{a \cdot b^2} $$
Extracted form:
$$ a^2 \cdot \sqrt[3]{a \cdot b^2} $$
Example 4
With a = 2 and b = 5:
$$ \sqrt[3]{2^7 \cdot 5^2} $$
Rewrite the power:
$$ 2^7 = 2^3 \cdot 2^3 \cdot 2 $$
$$ \sqrt[3]{2^3 \cdot 2^3 \cdot 2 \cdot 5^2} $$
Extract the perfect cubes:
$$ 2 \cdot 2 \cdot \sqrt[3]{2 \cdot 5^2} $$
$$ 4 \cdot \sqrt[3]{50} $$
Example 5
Consider
$$ \sqrt[3]{a^7 + b^2} $$
No extraction is possible because the radicand is a sum rather than a product.
Example. $$ \sqrt[3]{2^7 + 5^2} $$ cannot be simplified using extraction rules.
Proof
Consider
$$ \sqrt[n]{a^m \cdot b} $$
If m<n, extraction is not possible.
If m≥n, two cases must be considered.
Case 1
If m is a multiple of n, write m = n·q.
$$ \sqrt[n]{a^m} \cdot \sqrt[n]{b} $$
$$ \sqrt[n]{a^{n \cdot q}} \cdot \sqrt[n]{b} $$
Apply the invariant property of radicals:
$$ \sqrt[\frac{n}{n}]{a^{\frac{n \cdot q}{n}}} \cdot \sqrt[n]{b} $$
$$ a^q \cdot \sqrt[n]{b} $$
Case 2
If m is not a multiple of n, divide m by n:
$$ m = n \cdot q + r $$
Apply the laws of exponents:
am = an \cdot q \cdot ar
$$ \sqrt[n]{a^{n \cdot q} \cdot a^r \cdot b} $$
$$ \sqrt[n]{a^{n \cdot q}} \cdot \sqrt[n]{a^r \cdot b} $$
Apply the invariant property:
$$ a^q \cdot \sqrt[n]{a^r \cdot b} $$
The same reasoning extends to more complex radicals.
