Problem 37 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. Write your answers with only positive exponents. Assume that all variables represent positive real numbers. $$\left(4 a^{-2} b^{7}\right)^{1 / 2} \cdot\left(2 a^{1 / 4} b^{3}\right)^{5}$$

Step-by-Step Solution

Answer

The simplified expression is \(64a^{1/4}b^{37/2}\).

Step 1: Simplify Each Expression Separately

Start by simplifying each expression inside the parentheses. For the first expression \( \left(4a^{-2}b^7\right)^{1/2} \), apply the exponent \(1/2\) to each term:\( 4^{1/2} = 2 \),\( (a^{-2})^{1/2} = a^{-2 \cdot 1/2} = a^{-1} \),\( (b^7)^{1/2} = b^{7/2} \).So the expression becomes \( 2a^{-1}b^{7/2} \).

Step 2: Simplify the Second Expression

Next, simplify \( \left(2a^{1/4}b^3\right)^5 \) by applying the exponent \( 5 \) to each term:\( 2^5 = 32 \),\( (a^{1/4})^5 = a^{1/4 \cdot 5} = a^{5/4} \),\( (b^3)^5 = b^{3 \cdot 5} = b^{15} \).This gives us \( 32a^{5/4}b^{15} \).

Step 3: Multiply the Simplified Expressions

Multiply the simplified expressions from steps 1 and 2: \( 2a^{-1}b^{7/2} \cdot 32a^{5/4}b^{15} \).Multiply the constants: \( 2 \times 32 = 64 \).For the variable \( a \), add the exponents:\[ 64a^{-1 + 5/4}b^{7/2 + 15}. \]

Step 4: Add and Simplify the Exponents

Add the exponents for \( a \):\( -1 + 5/4 = -4/4 + 5/4 = 1/4 \). Add the exponents for \( b \):\( 7/2 + 15 = 7/2 + 30/2 = 37/2 \).The expression now is \( 64a^{1/4}b^{37/2} \).

Step 5: Ensure Exponents Are Positive

All final exponents are positive, so we don't need to make any changes. The expression is already written with only positive exponents: \( 64a^{1/4}b^{37/2} \).

Key Concepts

Exponentiation

Exponentiation is a core concept in algebra where a number, known as the base, is raised to a power, called the exponent. It simply means multiplying the base by itself a certain number of times. For example, in the expression \(3^2\), 3 is the base, and 2 is the exponent, indicating that 3 is multiplied by itself to yield \(3 imes 3 = 9\). When dealing with exponents applied to products of variables, each component inside the parentheses must be raised to the specified power. This is evident from the expression \((4a^{-2}b^7)^{1/2}\), where each term is individually raised to the 1/2 power, resulting in \(4^{1/2},\) \((a^{-2})^{1/2},\) and \((b^7)^{1/2}\). Remember:- Raising a term to an exponent means multiplying it by itself as many times as the number of the exponent indicates.- Distribute the exponent to each factor inside the parentheses for complete simplification.

Simplifying Expressions

Simplifying expressions involves using algebraic rules to transform complex expressions into a simpler or more workable form. It not only involves operations like addition and multiplication, but also applying laws of exponents to simplify terms.In practice, you simplify each component separately, as seen in the solution where \((4a^{-2}b^7)^{1/2}\) and \((2a^{1/4}b^3)^{5}\) are individually simplified by applying their respective exponents.Here’s how simplification works:

• Apply the power to each term: Break down the expression term-by-term under the operation, e.g., \((a^{1/4})^5 = a^{5/4}\).
• Combine like terms: When multiplying, add their exponents as with \(a^{-1} \cdot a^{5/4}\), resulting in \(a^{1/4}\).

The process ensures results are expressed in the most concise and accurate form.

Positive Exponents

Positive exponents indicate how many times to multiply a number by itself. Unlike negative exponents that imply division or reciprocation, positive exponents simplify the computational process to multiplication. When you simplify expressions, converting to positive exponents is often a requirement to make the answers more clear and standardized. For instance, when simplifying \(a^{-1}\), it can be rewritten with a positive exponent by interpreting it as \(1/a\), but it is often combined with other parts of the expression to yield entirely positive powers.Key points to remember about positive exponents:

• They indicate straightforward multiplication.
• Using them avoids complex division from negative exponents.
• Ensure your final expression uses only positive exponents for clarity.

This ensures your final answer is clear and conforms to mathematical conventions.