Problem 38 - Beginning and Intermediate Algebra

Question

Simplify. Assume all variables represent nonzero real numbers. The answer should not contain negative exponents. $$2\left(-3 a^{8} b\right)^{3}$$

Step-by-Step Solution

Answer

The simplified expression is \(-54 a^{24} b^3\).

Step 1: Identify the factors

The given expression is: \[ 2(-3a^8b)^3 \] This expression consists of the following factors: - a constant \(2\) - a power of an expression \((-3a^8b)^3\) Step 2: Apply the properties of exponents

Step 2: Simplify the power of an expression

When we raise the product of factors to an exponent, we can apply the exponent to each factor inside the parentheses: \[ (-3a^8b)^3 = (-3)^3(a^8)^3(b)^3 \] Step 3: Evaluate the constant part

Step 3: Calculate the power of -3

It's important to recall that when an odd number is raised to an odd power, the result will also be odd and negative: \[ (-3)^3 = -27 \] Step 4: Apply the properties of exponents (continued)

Step 4: Simplify the power of a and b

Apply the properties of exponents to the powers of a and b: \[ (a^8)^3 = a^{8 \cdot 3} = a^{24} \] \[ (b)^3 = b^{1 \cdot 3} = b^3 \] Step 5: Combine like terms and simplify

Step 5: Combine the constant, the power of a, and the power of b

Multiply the simplified factors obtained in steps 3 and 4: \[ 2 (-3a^8b)^3 = 2 \cdot -27 a^{24} b^3 = -54 a^{24} b^3 \] So, the simplified expression is: \[ -54 a^{24} b^3 \]

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial in simplifying expressions. Exponents, sometimes called "powers," describe how many times a number, known as the base, is multiplied by itself. The key properties of exponents help us perform operations with exponential expressions efficiently.

• **Product of Powers:** When multiplying two expressions with the same base, you add the exponents. For example, \( a^m \times a^n = a^{m+n} \).
• **Power of a Power:** When raising a power to another power, you multiply the exponents, such as \( (a^m)^n = a^{m \cdot n} \).
• **Power of a Product:** When a product is raised to a power, each factor in the product is raised to that power, seen in expressions like \((ab)^n = a^n b^n \).
• **Quotient of Powers:** When dividing two expressions with the same base, you subtract the exponents, shown as \( \frac{a^m}{a^n} = a^{m-n} \).
• **Zero Exponent:** Any base except zero raised to the zero power is one, for instance, \( a^0 = 1 \).

Applying these rules correctly can drastically simplify complex algebraic expressions.

Exponent Rules

Knowing the specific rules of exponents empowers you to simplify and solve expressions effectively, as seen in the exercise. These rules provide a clear guideline on how to handle bases and their powers.

• **Negative Exponents Rule:** A negative exponent means division or taking the reciprocal, \( a^{-n} = \frac{1}{a^n} \).
• **Fractional Exponents Rule:** Fractional exponents denote roots, where \( a^{\frac{m}{n}} = \sqrt[n]{a^m} \).
• **Distributive Rule for Exponents Over Multiplication:** This states you can distribute the exponent across the product, as in the case of \( (ab)^n = a^n b^n \).

When you encounter an expression like \((-3a^8b)^3\), applying the power of a product rule allows you to individually raise each component inside the bracket to the power of 3. Mastering these rules gives you the skill to manipulate and simplify expressions confidently.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form while maintaining the same value. The goal is to make the problem easier to work with by minimizing the complexity.Key steps in simplification:

• **Identifying Components:** Break down the expression into smaller components or factors. Determine if there are any common bases or terms you can work with.
• **Apply Exponent Rules:** Utilize rules such as power of a power and distributing exponents through products to simplify complex terms.
• **Perform Arithmetic Operations:** Calculate constants and coefficients, like \((-3)^3\) becoming \(-27\).
• **Recombine Terms:** After simplifying individual components, recombine them by following the order of operations, as you saw with \(2 \cdot (-27) a^{24} b^3 = -54 a^{24} b^3\).

These steps, applied in order, streamline the process of simplifying algebraic expressions, ensuring accuracy while enhancing efficiency.