Simplify cube root of (5a^2b)^6

Solution:

Step 1: Simplify the given expression.

• Step 1.1: Utilize the product rule on $5a^2b$.
  $\sqrt[3]{(5a^2b)^6} = \sqrt[3]{(5a^2)^6b^6}$

• Step 1.2: Apply the product rule to $5a^2$.
  $\sqrt[3]{(5a^2)^6b^6} = \sqrt[3]{5^6(a^2)^6b^6}$

• Step 1.3: Calculate $5^6$.
  $\sqrt[3]{5^6(a^2)^6b^6} = \sqrt[3]{15625(a^2)^6b^6}$

• Step 1.4: Handle the exponentiation in $(a^2)^6$.

  • Step 1.4.1: Apply the power of a power rule, $(a^m)^n = a^{mn}$.
    $\sqrt[3]{15625(a^2)^6b^6} = \sqrt[3]{15625a^{2\cdot6}b^6}$

  • Step 1.4.2: Multiply the exponents, $2 \times 6$.
    $\sqrt[3]{15625a^{12}b^6}$

Step 2: Express $15625a^{12}b^6$ as $(25a^4b^2)^3$.

$\sqrt[3]{15625a^{12}b^6} = \sqrt[3]{(25a^4b^2)^3}$

Step 3: Extract terms from under the cube root, assuming all variables represent real numbers.

$25a^4b^2$

Knowledge Notes:

To solve the given problem, we use several algebraic rules and properties:

1. Product Rule of Exponents: When multiplying two powers with the same base, you add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.

2. Power of a Power Rule: When raising a power to another power, you multiply the exponents. For example, $(a^m)^n = a^{mn}$.

3. Cube Root Simplification: The cube root of a variable raised to a power is the variable raised to the power divided by 3. For example, $\sqrt[3]{a^3} = a$.

4. Exponentiation of Numbers: Calculating the power of a number, such as $5^6$, which is $5 \times 5 \times 5 \times 5 \times 5 \times 5 = 15625$.

5. Simplification of Radical Expressions: When the exponent of the variable under the radical is divisible by the index of the radical, the variable can be taken out of the radical. For example, $\sqrt[3]{a^3} = a$.

In this problem, we applied these rules to simplify the cube root of the expression $(5a^2b)^6$. We first expanded the expression using the product rule, then simplified the numbers and exponents, and finally extracted the terms from under the cube root to get the simplified expression.