Problem 87 - Elementary and Intermediate Algebra

Question

Simplify each expression. Write the answers without negative exponents. All variables represent positive real numbers. See Example 8. $$ \frac{a^{3 / 4} a^{3 / 4}}{a^{1 / 2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( a \).

Step 1: Simplify the Numerator

In the expression \( \frac{a^{3/4} a^{3/4}}{a^{1/2}} \), start by simplifying the numerator. Use the property of exponents \( a^m \cdot a^n = a^{m+n} \). Therefore, \( a^{3/4} a^{3/4} = a^{3/4 + 3/4} = a^{6/4} \).

Step 2: Simplify the Exponent in the Numerator

Further simplify the exponent in the numerator from \( a^{6/4} \) to \( a^{3/2} \) by dividing both numerator and denominator of the fractional exponent by 2.

Step 3: Apply the Quotient of Powers Property

Now simplify the expression \( \frac{a^{3/2}}{a^{1/2}} \) using the quotient of powers property, \( \frac{a^m}{a^n} = a^{m-n} \). Thus, \( \frac{a^{3/2}}{a^{1/2}} = a^{3/2 - 1/2} = a^{2/2} \).

Step 4: Simplify the Exponent to a Whole Number

Now, simplify the expression \( a^{2/2} \) to \( a^1 \) since \( 2/2 = 1 \). Therefore, the final simplified expression is \( a \).

Key Concepts

Understanding the Laws of Exponents

In mathematics, exponents provide a shorthand for expressing repeated multiplication of the same number. The laws of exponents summarize important rules to make calculations more manageable. These laws allow us to simplify expressions efficiently, especially when dealing with multiple terms raised to various powers.

Here are some key laws:

• Product of Powers: This rule states that when multiplying two exponents with the same base, you can add the exponents, expressed as \( a^m \times a^n = a^{m+n} \).


• Quotient of Powers: When dividing two exponents with the same base, subtract the exponent in the denominator from the exponent in the numerator: \( \frac{a^m}{a^n} = a^{m-n} \).


• Power of a Power: To raise a power to another power, multiply the exponents together: \( (a^m)^n = a^{m\cdot n} \).

Applying these laws correctly is essential to simplify complex expressions effectively and avoid pitfalls in calculations.

Exploring Rational Exponents

Rational exponents are a way to express roots and powers in a unified form using fractions as exponents. These can often seem daunting but are quite straightforward once you understand the basic rule that governs them.

A rational exponent \( \frac{m}{n} \) means the base is both raised to the power of \( m \) and then the \( n \)th root is taken. In other words:

• \( a^{\frac{m}{n}} = \sqrt[n]{a^m} \).

Using rational exponents, one can easily convert between radical expressions and powers, making it more seamless to use the laws of exponents discussed earlier.

In the original exercise, rational exponents like \( a^{3/4} \) and \( a^{1/2} \) were simplified by combining and subtracting them appropriately using these concepts, illustrating their utility in simplifying expressions.

Simplifying Expressions Using Exponential Laws

Simplifying expressions involves using the laws of exponents to reduce complex terms into simpler, more manageable forms. This process aids in solving equations efficiently and clearing the path to finding a final answer or solution.

To illustrate simplification, consider the expression \( \frac{a^{3/4} a^{3/4}}{a^{1/2}} \). Follow these steps:

• Step 1: Simplify the numerator using the Product of Powers law, where \( a^{3/4} \times a^{3/4} = a^{6/4} \).
• Step 2: Convert \( a^{6/4} \) to \( a^{3/2} \) by simplifying the fraction.
• Step 3: Apply the Quotient of Powers law to divide \( a^{3/2} \) by \( a^{1/2} \), resulting in \( a^{1} \), as \( 3/2 - 1/2 = 1 \).

Expressing the final answer without negative exponents is a crucial part of simplification, ensuring clarity and precision in mathematical communication. Simplifying expressions is not about removing complexity but rather expressing it in its most understandable form.