Problem 36 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ \left(2^{3} \cdot 3^{3}\right)^{1 / 3} $$

Step-by-Step Solution

Answer

The solution to the expression \(\left(2^{3} \cdot 3^{3}\right)^{1 / 3}\) is 6.

Step 1: Identify innermost operation

In this exercise, the expression inside the paranthesis is the inner operation. Thus, first follow the arithmetic rule of completing operations inside parentheses. Within that, identify the problem of form \(a^{m} \cdot a^{n}\), which uses the law of exponents allowing to add the exponents with the same base. Here, the base for both terms inside the parantheses is different, so this law doesn't apply.

Step 2: Calculate the terms within the parenthesis

The expression inside the parantheses is \(2^{3} \cdot 3^{3}\). This can be broken down into two parts, \(2^{3} = 8\) and \(3^{3} = 27\). Multiply these two results: \(8 \cdot 27 = 216\)

Step 3: Apply the power-of-a-power rule

The next operation is to raise the result of Step 2 to the power \(1 / 3\), which follows the form \(a^{mn}\). This uses the power-of-a-power rule. In this case, this rule allows us to take the cube root of 216, effectively simplifying the expression to \(216^{1 / 3}\)

Step 4: Simplify further

Solving the expression \(216^{1 / 3}\) gets the cube root of 216. Simplify this to find it equals 6.

Key Concepts

Exponents and Powers

Exponents and powers are fundamental concepts in mathematics that help us express repeated multiplication concisely. When you have a number like 2 raised to the power of 3, written as \(2^3\), it means you multiply 2 by itself three times: \(2 \times 2 \times 2 = 8\). This compact form not only makes calculations simpler, but also helps in handling large numbers efficiently.

Key points about exponents include:

• The base is the number being multiplied (e.g., 2 in \(2^3\)).
• The exponent (3 in \(2^3\)) indicates how many times the base is used as a factor.
• Powers of numbers make it easy to represent and compute large products.

Understanding exponents is useful as they appear frequently in mathematical expressions, and recognizing patterns within exponents can simplify complex calculations.

Laws of Exponents

The laws of exponents play a crucial role in simplifying expressions involving powers. They allow us to manipulate and break down expressions into simpler forms. Here are some important laws you'll often use:

• Product of powers: For the same base, \(a^m \times a^n = a^{m+n}\).
• Power of a power: \((a^m)^n = a^{m \cdot n}\).
• Power of a product: \((a \cdot b)^n = a^n \cdot b^n\).
• Quotient of powers: \(a^m \div a^n = a^{m-n}\) when \(a eq 0\).
• Zero as an exponent: Any non-zero number \(a\) raised to the power of 0 is 1, i.e., \(a^0 = 1\).

These rules help you simplify calculations, allowing you to focus on the core of the problem rather than getting bogged down by numbers. They are especially handy in algebra when working with variables and exponents. Understanding these laws empowers you to manipulate and simplify mathematical expressions effectively.

Cube Roots

Finding the cube root is the reverse operation of cubing a number. If squaring a number means multiplying it by itself, cubing means multiplying it by itself twice. Thus, finding a cube root means identifying a number which, when used in three-fold multiplication, yields the given number.

For example, if you have \(216\), its cube root is the number \(6\) because \(6 \times 6 \times 6 = 216\). The notation for a cube root is \(\sqrt[3]{\ }\), so the cube root of 216 is written as \(\sqrt[3]{216} = 6\).

• Numerical relations: Just as squares and square roots are related, cubes and cube roots work oppositely to undo each other.
• Applications: Cube roots are useful in geometry, particularly when dealing with volumes as they appear naturally when working with three-dimensional measurements.
• Expressions: Often, cube roots help simplify complex expressions by reducing exponents or solving equations involving cubic terms.

Knowing how to find and use cube roots is helpful in understanding the nature of numbers and is essential for solving problems in both arithmetic and algebra.