Problem 50 - Intermediate Algebra

Question

Square or cube each quantity and simplify the result. $$ (-3 \sqrt[3]{10 y^{3}})^{3} $$

Step-by-Step Solution

Answer

The simplification of \(-3 \sqrt[3]{10 y^{3}})^{3} is \(-270 y^{3}\).

Step 1: Distribute the Exponent

The expression is \[ (-3 \sqrt[3]{10 y^{3}})^{3} \]. We need to distribute the exponent of 3 to each component inside the parentheses separately. That is, we write it as:\[ (-3)^{3} \times (\sqrt[3]{10 y^{3}})^{3} \].

Step 2: Calculate the Cube of -3

Calculate \[ (-3)^{3} \]. Since, \[ (-3) \times (-3) \times (-3) = -27 \], thus \[ (-3)^{3} = -27 \].

Step 3: Simplify the Cube of the Cube Root

Now, consider the component \[ (\sqrt[3]{10 y^{3}})^{3} \]. The cube of the cube root is simply the original quantity because \[ (a^{1/3})^{3} = a \] for any \( a \). Hence, \[ (\sqrt[3]{10 y^{3}})^{3} = 10 y^{3}. \]

Step 4: Multiply the Simplified Components

Finally, multiply the results from the previous steps: \[ -27 \times 10 y^{3} = -270 y^{3}. \] Therefore, \[ (-3 \sqrt[3]{10 y^{3}})^{3} = -270 y^{3}. \]

Key Concepts

Understanding Exponents

Exponents are a fundamental concept in math that help us easily express repeated multiplication. Instead of writing a number multiple times, we can use a base and an exponent. For instance, in the expression \((-3)^3\), \(-3\) is the base and \(3\) is the exponent. This means we multiply \(-3\) three times: \((-3) \times (-3) \times (-3) = -27\).

• The base tells us which number to multiply.
• The exponent tells us how many times to multiply the base by itself.

Identifying and working with exponents can initially seem tricky, but remembering that they simplify repeated multiplication can make them more approachable. It's like a shortcut that tells how powerful a number becomes. When looking at equations or expressions, first identify where exponents are involved and break down the expression step-by-step. This approach will reveal the underlying simplicity of expressions with exponents.

Exploring Cube Roots

Cube roots work as the opposite of cubing a number. If you see a cube root, like \( \sqrt[3]{x} \), you're looking for a number that, when multiplied by itself three times, equals \(x\). For example, if you have \( \sqrt[3]{8} \), the answer is \(2\) since \(2 \times 2 \times 2 = 8\).

• Cubing a number and taking the cube root are inverse operations.
• Cube roots can simplify expressions involving cube powers.

In the given problem, simplifying \( (\sqrt[3]{10 y^{3}})^{3} \) is straightforward. Using the property \((a^{1/3})^3 = a\), this expression simplifies to \(10 y^3\). Understanding the inverse relationship will help you manage and simplify terms that involve exponents and roots. Always look for opportunities where a root and an exponent can balance each other out to make simplification much clearer.

Multiplying Integers

Multiplication of integers involves basic arithmetic rules observed with all whole numbers. When multiplying integers, it is crucial to pay attention to the signs of the numbers involved.

• Multiplying two positive numbers or two negative numbers results in a positive product.
• Multiplying a positive and a negative number gives a negative product.

In our problem, after determining that \((-3)^3 = -27\), we need to multiply \(-27\) by \(10 y^3\) from the cube root simplification. The process is straightforward. First, multiply the numerical coefficients: \(-27 \times 10 = -270\). Next, maintain the variable part, resulting in \(-270 y^3\). Observing the rule about the signs helps prevent errors, as managing signs correctly is key to solving multiplication of integers correctly.