Problem 30 - Algebra 2

Question

Simplify. $$ \sqrt[3]{(2 z)^{6}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(4z^{2}\).

Step 1: Understand the problem

We need to simplify the expression \( \sqrt[3]{(2z)^{6}} \). This involves dealing with the cube root and the power of the expression \((2z)^{6}\).

Step 2: Apply the properties of exponents

Using the property \((a^m)^n = a^{m \cdot n}\), rewrite the expression inside the cube root as \[(2z)^{6} = 2^{6} \cdot z^{6}\]

Step 3: Apply the cube root

Now take the cube root of the expression, using the property \( \sqrt[3]{a^n} = a^{n/3} \), apply this to both components:\[\sqrt[3]{(2^{6} \cdot z^{6})} = \sqrt[3]{2^{6}} \cdot \sqrt[3]{z^{6}}\]

Step 4: Simplify each component under the cube root

For the cube root of \(2^{6}\), \[\sqrt[3]{2^{6}} = 2^{6/3} = 2^{2} = 4\]For the cube root of \(z^{6}\),\[\sqrt[3]{z^{6}} = z^{6/3} = z^{2}\]

Step 5: Combine the simplified results

Multiply the simplified results from Step 4:\[4 \cdot z^{2} = 4z^{2}\]

Key Concepts

Cube Roots

Cube roots are mathematical operations that determine which number, when cubed (multiplied by itself twice), gives you the original number. In simpler terms, the cube root of a number \( x \) is the number \( y \) such that \( y^3 = x \). The cube root is denoted by the radical symbol with an index of 3, written as \( \sqrt[3]{x} \).
In our example, we take the cube root of a power term \((2z)^6\). This involves the property \( \sqrt[3]{a^n} = a^{n/3} \), where the exponent is divided by 3. These operations are critical in simplifying expressions involving cube roots.
Always look for perfect cube components inside the radical for easier computation, as cube roots often simplify neatly, making calculations less complex.

Properties of Exponents

Understanding the properties of exponents is crucial for simplifying algebraic expressions, especially those involving powers and roots. An essential property is \((a^m)^n = a^{m \cdot n}\). This allows us to transform powers raised to another power into a single power expression. For example, \((2z)^6\) can be rewritten as \(2^6 \cdot z^6\).
Another vital property is the division or root of exponents, given by \(a^{m/n} = \sqrt[n]{a^m}\) or \(a^{m/n} = (a^m)^{1/n}\). This helps when applying roots to powers, effectively simplifying the process to basic multiplication and division operations.

• Power of a product: \((ab)^m = a^m \cdot b^m\)


• Power of a quotient: \((\frac{a}{b})^m = \frac{a^m}{b^m}\)


• Zero exponent: \(a^0 = 1\) (provided \(a eq 0\))


• Negative exponents: \(a^{-m} = \frac{1}{a^m}\)

Utilizing these properties correctly allows for breaking down complex expressions into more manageable parts, especially when combining different operations like multiplication and division with powers.

Simplification of Algebraic Expressions

Simplifying algebraic expressions means making them easier to work with or understand without changing their value. This involves reducing the complexity of expressions, which can sometimes be daunting, especially with multiple operations.
In the provided exercise, the expression \( \sqrt[3]{(2z)^6} \) was simplified by applying properties of exponents and cube roots. By first expanding the expression \((2z)^6\) to \(2^6 \cdot z^6\), and then applying the cube root to each part separately, we could achieve a simplified form: \(4z^2\).
The process includes:

• Identifying and applying suitable rules, such as power simplification and root extraction.


• Reducing terms wherever possible, turning multiplication into exponent operations when needed.


• Ensuring each step progressively simplifies the expression, checking consistency with initial values.

By following these approaches, algebraic simplification not only becomes feasible but transforms complex equations into understandable and solvable forms, enhancing problem-solving efficiency.