Problem 39 - Algebra and Trigonometry

Question

\(29-46\) Simplify each expression. $$ \left(2 y^{2}\right)^{3} $$

Step-by-Step Solution

Answer

\((2y^2)^3 = 8y^6\).

Step 1: Understand the Expression

The expression we need to simplify is \((2y^2)^3\). This involves an exponential term where a term inside the parentheses is raised to the power of 3.

Step 2: Apply the Power of a Product Property

To simplify \((2y^2)^3\), use the "power of a product" rule, which states that \((ab)^n = a^n \cdot b^n\). Here, \(a = 2\) and \(b = y^2\), so \((2y^2)^3 = 2^3 \cdot (y^2)^3\).

Step 3: Simplify Each Component

Calculate \(2^3\) which is 8. For \((y^2)^3\), use the "power of a power" rule, which states \((a^m)^n = a^{m \cdot n}\). So, \((y^2)^3 = y^{2 \cdot 3} = y^6\).

Step 4: Combine the Simplified Terms

Now combine the terms from the previous step: \(2^3 \cdot (y^2)^3 = 8 \cdot y^6\). So, the expression \((2y^2)^3\) simplifies to \(8y^6\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, letters (variables), operation symbols, and sometimes parentheses. They represent mathematical concepts and can be simplified or solved. In our example,

• The expression \((2y^2)^3\) is an algebraic expression.
• It contains numbers, which are constants like 2, and a variable, \(y^2\), which changes its value.
• The goal is often to simplify these expressions.

Algebraic expressions are the building blocks of algebra, allowing us to write general formulas and solve problems.Learning to recognize and work with these expressions is a crucial first step in mastering mathematics.

Power of a Power Rule

The power of a power rule is a fundamental concept in exponentiation. It helps simplify expressions where an exponential term is raised to another power. This rule can be expressed as:

• \((a^m)^n = a^{m \cdot n}\),

where \(a\) is the base, and \(m\) and \(n\) are the exponents. In our exercise:

• The term \((y^2)^3\) uses this rule.
• Multiply the exponents 2 and 3 to get \(y^6\).

This rule simplifies the calculation process by combining exponents, helping manage large numbers efficiently.

Power of a Product Rule

The power of a product rule is another essential property used in exponentiation. It simplifies expressions where a product of factors is raised to a power. According to this rule:

• \((ab)^n = a^n \cdot b^n\),

where \(a\) and \(b\) are factors and \(n\) is the common exponent. In the example given:

• \((2y^2)^3\) involves multiplying both the number 2 and the term \(y^2\) raised to the power of 3.
• The expression simplifies as follows: \(2^3 \cdot (y^2)^3\).
• This gives us \(8 \cdot y^6\).

Recognizing and applying the power of a product rule helps you to efficiently break down complex expressions.