Problem 95 - Algebra A Combined Function

Question

Simplify each exponential expression. $$ \left(-2 x^{3} y^{2}\right)^{5} $$

Step-by-Step Solution

Answer

The simplified expression is \(-32 x^{15} y^{10}\).

Step 1: Apply the Power to Each Factor

The expression \( \left(-2 x^{3} y^{2}\right)^{5} \) can be expanded by applying the power of 5 to each of the factors inside the parentheses. This means we distribute the exponent to the constant and each variable: \((-2)^5 (x^{3})^5 (y^{2})^5\).

Step 2: Simplify the Constant

Calculate \((-2)^5\). A negative number raised to an odd exponent remains negative. So, \((-2)^5 = -32\).

Step 3: Simplify the Exponent of the First Variable

Apply the power of a power rule for exponents: \((x^{3})^5 = x^{3 \times 5} = x^{15}\).

Step 4: Simplify the Exponent of the Second Variable

Use the same power of a power rule for the second variable: \((y^{2})^5 = y^{2 \times 5} = y^{10}\).

Step 5: Combine the Results

Now combine all of the simplified parts together: \(-32 x^{15} y^{10}\).

Key Concepts

Power of a Power Rule

The "power of a power rule" is a fundamental principle in algebra that helps us manage complex exponential expressions. When we see an expression like \((x^a)^b\), this rule tells us to multiply the exponents: \(x^{a imes b}\). This concept becomes particularly useful when dealing with variables raised to an exponent within a larger exponential expression. Applying this rule simplifies our calculations, making it easier to evaluate expressions like \((x^{3})^5\), giving us \(x^{15}\). Remember this rule whenever you're faced with exponents piled on top of each other. It clarifies calculations and reduces potential errors in solving algebraic problems.

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra, enabling us to present equations in their most concise form. The goal is to make the expression easier to understand and work with. Consider the expression \((-2 x^{3} y^{2})^5\). By simplifying, we rewrite and solve each part of the expression separately:

• The constant \((-2)^5\) is calculated as \(-32\).
• The first variable part \((x^3)^5\) is simplified to \(x^{15}\).
• The second variable \((y^2)^5\) simplifies to \(y^{10}\).

Combining these pieces gives us the simplified expression \(-32 x^{15} y^{10}\). By breaking down each component, we can handle even complex expressions efficiently.

Exponents and Coefficients

Understanding exponents and coefficients is essential for mastering algebraic expressions. Exponents tell us how many times a number, known as the base, is multiplied by itself. For example, in \(x^3\), the base \(x\) is multiplied by itself three times. Coefficients, on the other hand, are the numbers placed in front of variables, showing how many times the term is added together. In \(5x^3\), "5" is the coefficient, indicating that \(x^3\) is five times as large. When simplifying expressions, handle the coefficients and exponents with care. For instance, in the example \((-2 x^{3} y^{2})^5\), the coefficient is \(-2\) and the exponents are on the variables \(x\) and \(y\). Applying powers correctly involves distributing the exponent to both coefficients and variables, ensuring accurate calculations. This results in the simplified term \(-32 x^{15} y^{10}\). Understanding these components helps in creating accurate and compliant algebraic solutions.