Problem 98 - Intermediate Algebra

Question

Simplify each exponential expression. $$ \frac{\left(2 a^{-1} b^{2}\right)^{3}}{\left(8 a^{2} b\right)^{-2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(512ab^8\).

Step 1: Apply the Power of a Power Rule

First, apply the power of a power rule to the numerator \((2a^{-1}b^2)^3\). This rule states \((x^m)^n = x^{m imes n}\). Therefore, \((2a^{-1}b^2)^3 = 2^3 \cdot a^{-1 \times 3} \cdot b^{2 \times 3}\). Simplifying gives:\[\frac{8}{a^3} \cdot b^6\].

Step 2: Simplify the Denominator

Next, simplify the denominator \((8a^2b)^{-2}\) using the power of a power rule:\((x^m)^n = x^{m \times n}\). Therefore, \( (8a^2b)^{-2} = 8^{-2} \cdot a^{2 \times (-2)} \cdot b^{-2}\). Simplifying gives \[\frac{1}{64a^4b^2}\].

Step 3: Simplify the Entire Fraction

We now have \[\frac{8a^{-3}b^6}{\frac{1}{64a^4b^2}}\]. To simplify, apply the rule \(\frac{x}{y^{-1}} = xy\), leading to:\[8a^{-3}b^6 \cdot 64a^4b^2\].

Step 4: Combine Exponents Using Product Rule

Apply the product rule \(x^m \cdot x^n = x^{m+n}\) to combine similar bases:\[8 \cdot 64 \cdot a^{-3 + 4} \cdot b^{6 + 2}\]. Simplify further to obtain \[512a^1b^8\].

Step 5: Simplify the Constants

Simplifying the coefficient, we simply multiply: \(8 \times 64 = 512\). The expression becomes \(512a^1b^8\) or simply \[512ab^8\].

Key Concepts

Power of a Power Rule

The power of a power rule is an essential concept when working with exponential expressions. It allows us to simplify expressions where an exponent is raised to another exponent. The rule is expressed as \((x^m)^n = x^{m \times n}\). Applying this rule, we simplify the expression by multiplying the exponents. For example, using the power of a power rule on the expression \((2a^{-1}b^2)^3\) results in:

• \(2^{3} = 8\)
• \(a^{-1 \times 3} = a^{-3}\)
• \(b^{2 \times 3} = b^{6}\)

Applying this rule effectively simplifies the expression from a more complex form into one that is easier to understand and work further with.

Simplifying Fractions

Simplifying fractions with exponents involves making the expression more manageable by reducing it to its simplest form. This process requires applying rules of exponents and occasionally converting between different equivalents of expressions. For example, to simplify the fraction \(\frac{8a^{-3}b^6}{\frac{1}{64a^4b^2}}\), a reciprocal rule is used to "invert" the denominator, which turns division into multiplication, giving us:

• Multiply numerator by reciprocal of denominator
• \(8a^{-3}b^6 \cdot 64a^4b^2\)

This simplifies the expression into a more straightforward calculation, making further simplification steps easier. Remember, simplifying fractions properly sets the stage for correctly applying other rules as seen later in this process.

Product Rule

The product rule, another fundamental principle of exponent operations, involves combining bases of the same type by adding their exponents. The rule states that \(x^m \cdot x^n = x^{m+n}\). This is used when combining exponents during simplification processes, especially after resolving the initial expression to like bases.
Utilizing our example, when faced with the expression \(8 \cdot 64 \cdot a^{-3 + 4} \cdot b^{6 + 2}\), apply the product rule like so:

• The similar bases with \(a\) become \(a^{1}\)
• The similar bases with \(b\) become \(b^{8}\)
• The constants are multiplied normally (\(8 \times 64 = 512\))

By using the product rule, the complex expression is expertly simplified, showing how different elements interact within exponential expressions.

Simplifying Exponents

Simplifying exponents is often the final stage in refining exponential expressions. It consists of reducing all components to their simplest terms while ensuring numerical coherence throughout the expression. This involves understanding and combining previous steps effectively, yielding a compact form.
In the exercise, after applying the power of a power rule and product rule, we are left with \(512a^1b^8\). Here are the final thoughts to keep in mind for simplifying exponents:

• Ensure all exponents are combined or simplified, as with \(a^1\) to \(a\)
• Verify numerical coefficients are evaluated appropriately, leading to \(512\)

The ultimate expression, \(512ab^8\), reflects the resolved form which showcases the elegance and effectiveness of combining exponential rules to simplify challenging problems.