Problem 100 - Algebra A Combined Function

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

Start by applying the power of a power rule, which states that \( (x^m)^n = x^{m \cdot n} \). For the expression \(\left(2a^{-1}b^{2}\right)^3\), it becomes: \[ \left(2a^{-1}b^{2}\right)^3 = 2^3 (a^{-1})^3 (b^2)^3 \ = 8a^{-3}b^6 \]

Step 2: Simplify the Denominator

For the denominator, \(\left(8 a^2 b\right)^{-2}\), apply the power of a power rule again: \[\left(8 a^2 b\right)^{-2} = 8^{-2}(a^2)^{-2}(b)^{-2} = \frac{1}{8^2} \cdot a^{-4} \cdot b^{-2} = \frac{1}{64}a^{-4}b^{-2}\\]

Step 3: Divide the Exponents

Combine the results from the numerator and denominator into a single expression: \[\frac{8a^{-3}b^6}{\frac{1}{64}a^{-4}b^{-2}} \] This expression simplifies by multiplying by the reciprocal of the denominator: \[= 8a^{-3}b^6 \times 64a^4b^2 = 512a^{(4 - 3)}b^{(6 + 2)}\]

Step 4: Final Simplification

Finally, simplify the exponents: \[ 512a^{1}b^{8} = 512ab^{8} \]

Step 5: Write the Final Answer

The simplified form of the given exponential expression is: \[ 512ab^{8} \]

Key Concepts

Power of a Power Rule

When dealing with exponential expressions, one of the most useful tools is the Power of a Power Rule. This rule helps simplify complex expressions and is quite straightforward: if you have a power raised to another power, you multiply the exponents together. Imagine having an expression \( (x^m)^n \). By using the power of a power rule, it becomes \( x^{m \cdot n} \).
This rule is particularly handy when you're simplifying terms like \( (2a^{-1}b^2)^3 \), as shown in our example. Here, each part of the term inside the parentheses is raised to the third power:

• \( 2^3 \), which simply equals 8.
• \( (a^{-1})^3 \), becomes \( a^{-3} \) after multiplying the exponents (\( -1 \cdot 3 \)).
• \( (b^2)^3 \), simplifies to \( b^6 \) by multiplying the exponents (\( 2 \cdot 3 \)).

Putting it all together, the use of the Power of a Power Rule helps quickly reduce complex exponentiation into simpler pieces which are then much easier to manage in further calculations.

Exponentiation

Exponentiation is a method involving repeated multiplication of a number by itself. It’s identified by a small raised number, called the exponent, which denotes how many times to multiply the base by itself. Let's take a closer look.
For instance, in the expression \( 2^3 \), 2 is the base and 3 is the exponent, which means you multiply 2 by itself three times: \( 2 \times 2 \times 2 = 8 \). This principle applies equally to variables: \( a^2 \) means \( a \times a \).
Understanding exponentiation is crucial when simplifying both the numerator and denominator in expressions like the one given in this problem. It allows you to expand and then compress the terms to make computation simpler.
Rules for different operations such as multiplying exponents (add the exponents) or raising a power to a power (multiply the exponents) are essential to master for further simplifying complex algebraic expressions.

Simplifying Expressions

The process of simplifying expressions is an important skill in algebra that involves writing expressions in the most compact and understandable form. This typically involves eliminating parentheses, combining like terms, and reducing overall complexity without changing the value of the expression.
Let’s look at the example given. The expression \( \frac{8a^{-3}b^6}{\frac{1}{64}a^{-4}b^{-2}} \) utilized the concept of multiplying by a reciprocal to simplify. When you divide by a fraction, it is the same as multiplying by its reciprocal — for example, multiplying by \( 64 \) instead of dividing by \( \frac{1}{64} \).
By multiplying the components of the numerator and the reciprocal of the denominator, you combine like bases:

• For \( a \), the exponents \( -3 \) and \( 4 \) are added to get \( 1 \).
• For \( b \), the exponents \( 6 \) and \( 2 \) are added to get \( 8 \).

Finally, multiplying the constants gives \( 512 \).
This process requires understanding how to handle exponents and apply algebraic rules correctly to simplify expressions efficiently.