Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: Simplify. $$ \left(-2 x y^{2}\right)^{5}\left(\frac{x^{7}}{8 y^{3}}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(-4x^{12}y^7\).

Step 1: Simplify the Power Expression

First, simplify the expression \( (-2xy^2)^5 \) by applying the power rule \( (ab)^n = a^n \cdot b^n \). This gives:\[ (-2)^5 \cdot (x)^5 \cdot (y^2)^5 = -32x^5y^{10} \] because \( (-2)^5 = -32 \).

Step 2: Expand the Division Expression

Consider the expression \( \frac{x^7}{8y^3} \). This can be rewritten explicitly as:\[ \frac{x^7}{8y^3} = \frac{1}{8} \cdot x^7 \cdot y^{-3} \] interpreting the division as negative power for \( y \).

Step 3: Combine the Expressions

Now combine the results from Step 1 and Step 2. The expressions are:\[ -32x^5y^{10} \quad \text{and} \quad \frac{1}{8}x^7y^{-3} \]Multiply these expressions together:\[ -32x^5y^{10} \cdot \frac{1}{8}x^7y^{-3} = \left(-32 \cdot \frac{1}{8}\right) \cdot x^{5+7} \cdot y^{10-3} \] which becomes:\[ -4x^{12}y^7 \].

Step 4: Simplify the Result

Finally, simplify as: \[ -4x^{12}y^7 \] as the final simplified expression result.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving numbers called the base and the exponent or power. It represents repeated multiplication of the base. For instance, in the expression \((-2xy^2)^5\), \(-2xy^2\) is the base and 5 is the exponent. This tells us to multiply \(-2xy^2\) by itself 5 times.

• The process of exponentiation can drastically transform an expression, especially when multiple variables are involved.
• Remember, each part of the base, such as \(-2\), \(x\), and \(y^2\), must be raised to the power independently, as shown by our solution where we obtained \((-2)^5\), \(x^5\), and \((y^2)^5\).
• This results in the expression \(-32x^5y^{10}\).

Simplification

Simplification is the process of reducing a mathematical expression into its simplest form. This is like cleaning up a room, making everything more organized and concise. By simplifying, we make expressions easier to understand or work with.

• In mathematics, specific rules and laws guide simplification, such as multiplying similar bases and adding their exponents.
• For example, the products of different terms, like \(-32x^5y^{10}\) from \((-2xy^2)^5\), are manipulated to remove unnecessary complexity.
• Thus, simplification transforms our problem into a more digestible solution, introducing more manageable numbers and consolidated terms.

Power Rule

The Power Rule in algebra is a fundamental principle that helps in solving expressions with exponents. It states that to raise a power to a power, you multiply the exponents.

Applying the Rule

This rule was directly applied in the expression \((y^2)^5\), where we took the exponent 2 and multiplied it by 5, resulting in \(y^{10}\). The same was done with \(x^5\) and \((-2)^5\). This helps streamline the process of working with complex expressions.

• The general formula for the power rule is \((a^m)^n = a^{m\cdot n}\).
• This enables swift calculation when the base contains variables or coefficients.
• As seen here, \(y^{10}\) came from \(y^2\) raised to the fifth power using the power rule.

Negative Exponents

Negative exponents represent the reciprocal of the base raised to the opposite positive exponent. A negative exponent indicates division rather than multiplication.

Reversing the Operation

The expression \(y^{-3}\) signifies \(\frac{1}{y^3}\). This occurs in our exercise when simplifying the term with division \(\frac{1}{8}x^7y^{-3}\). By acknowledging that a negative exponent such as “\(-n\)” flips the base into the denominator, the expression finds its simplest form.

• The rule to remember is \(a^{-n} = \frac{1}{a^n}\).
• This paves the way for rewriting division in terms of multiplication, significantly simplifying the process of solving and evaluating expressions.
• Such methods ease transformations between complex division scenarios into clearer multiplication terms.