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

The given problem is a mathematical expression that requires simplification. Specifically, you are asked to find the cube root of the entire expression (5a^2b)^6. The expression involves an exponent and a radical. The main aim would be to apply the properties of exponents and radicals to simplify the expression to a form that is more easily interpretable or possibly in its simplest algebraic form.

$\sqrt[3]{{((5 a^{2} b))}^{6}}$

Answer

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Solution:

Step 1: Simplify the given expression.

Step 1.1: Utilize the product rule on $5 a^{2} b$ .
$\sqrt[3]{(5 a^{2} b)^{6}} = \sqrt[3]{(5 a^{2})^{6} b^{6}}$
Step 1.2: Apply the product rule to $5 a^{2}$ .
$\sqrt[3]{(5 a^{2})^{6} b^{6}} = \sqrt[3]{5^{6} (a^{2})^{6} b^{6}}$
Step 1.3: Calculate $5^{6}$ .
$\sqrt[3]{5^{6} (a^{2})^{6} b^{6}} = \sqrt[3]{15625 (a^{2})^{6} b^{6}}$
Step 1.4: Handle the exponentiation in $(a^{2})^{6}$ .
- Step 1.4.1: Apply the power of a power rule, $(a^{m})^{n} = a^{m n}$ .
  $\sqrt[3]{15625 (a^{2})^{6} b^{6}} = \sqrt[3]{15625 a^{2 \cdot 6} b^{6}}$
- Step 1.4.2: Multiply the exponents, $2 \times 6$ .
  $\sqrt[3]{15625 a^{12} b^{6}}$

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

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

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

$25 a^{4} b^{2}$

Knowledge Notes:

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

Product Rule of Exponents: When multiplying two powers with the same base, you add the exponents. For example, $a^{m} \cdot a^{n} = a^{m + n}$ .
Power of a Power Rule: When raising a power to another power, you multiply the exponents. For example, $(a^{m})^{n} = a^{m n}$ .
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, $\sqrt[3]{a^{3}} = a$ .
Exponentiation of Numbers: Calculating the power of a number, such as $5^{6}$ , which is $5 \times 5 \times 5 \times 5 \times 5 \times 5 = 15625$ .
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 $(5 a^{2} b)^{6}$ . 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.