Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: Simplify. $$ \left(-3 a^{2} b^{-5}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(-27a^6b^{-15}\).

Step 1: Apply the Power of a Power Rule

First, use the power of a power rule, which states that \((x^m)^n = x^{m*n}\). Apply this rule to each component within the parentheses \\((-3a^2b^{-5})^3\):\ - The coefficient \(-3\) becomes \((-3)^3\). - The base \(a^2\) becomes \(a^{2*3}=a^6\). - The base \(b^{-5}\) becomes \(b^{-5*3}=b^{-15}\).

Step 2: Simplify Each Component

Calculate each part:- The coefficient: \((-3)^3 = -27\).- The \(a\) term: \(a^6\).- The \(b\) term: \(b^{-15}\).Thus, you have \-27a^6b^{-15}\.

Step 3: Final Expression

The simplified expression is a combination of all components: \[-27a^6b^{-15}\].This is the final simplified form of the original expression.

Key Concepts

Power of a Power Rule

When we encounter an expression like \((-3a^2b^{-5})^3\), we need to apply one of the most useful rules in algebra, the power of a power rule. This rule helps simplify expressions where exponents are themselves raised to another power. The rule states that

• \((x^m)^n = x^{m \cdot n}\)

This means that when you have a base raised to a power, which is then raised to another power, you multiply the exponents. For example, in the expression \((-3a^2b^{-5})^3\):- The coefficient \(-3\) is raised to the power of 3, resulting in \(-3)^3\). - The base \(a^2\) becomes \(a^{2 \cdot 3} = a^6\). - The base \(b^{-5}\) becomes \(b^{-5 \cdot 3} = b^{-15}\). This simple multiplication of exponents in this rule makes handling complex exponential expressions manageable.

Simplification

Simplification is the process of reducing expressions to their simplest form. It's like cleaning up your workspace—removing unnecessary clutter to see the main things. In the context of algebraic expressions, simplification involves performing operations according to mathematical rules to make the expression more concise and easier to understand.

Let's simplify the expression from our exercise. After applying the power of a power rule, we have:\(-3\) raised to the power of 3, and the other components expanded as mentioned earlier.

Now, calculate each of these:

• The coefficient \(-3\) becomes \((-3)^3 = -27\).
• \(a^2\) raised to the power 3 becomes \(a^6\).
• \(b^{-5}\) raised to the power 3 becomes \(b^{-15}\).

The final simplified expression is \(-27a^6b^{-15}\). In this process, we use basic mathematical operations to transform the expression into a more straightforward form, conveniently combining all components at the end.

Exponents

Exponents are a way of expressing repeated multiplication of a number by itself. For example, in the expression \(a^n\), \(a\) is the base, and \(n\) is the exponent. It tells us how many times the base is multiplied by itself:
\[a^n = a \cdot a \cdot a \cdots (\text{n times})\]\
Exponents are fundamental in math because they help simplify expressions involving repeated multiplication and make calculations much easier.

• Positive exponents indicate standard multiplication. For example, \(a^3 = a \cdot a \cdot a\).
• Negative exponents, as seen in our exercise with \(b^{-5}\), represent the reciprocal of the base raised to the positive exponent, \(b^{-5} = \frac{1}{b^5}\).

Using exponents effectively allows us to manage complex expressions by reducing them into smaller, more manageable pieces.

Negative Exponents

Negative exponents can seem tricky at first, but they follow a simple rule that turns them into a powerful tool in algebra. A negative exponent means that instead of multiplying, you're taking the reciprocal of the number. In simpler terms, it flips the base, turning it into a fraction.

The rule of negative exponents is written as:

• \(a^{-n} = \frac{1}{a^n}\)

This rule is handy when dealing with equations and expressions that need simplification. For instance, in our exercise expression, the term \(b^{-15}\) is equal to \(\frac{1}{b^{15}}\).

Negative exponents help when simplifying and rewriting expressions to a more standard form, clarifying the relationships between different parts of the expression. By understanding and applying negative exponents, we can express repeated division in an algebraically concise way.