Problem 51 - Algebra 1: Concepts and Skills

Question

Simplify the expression. Use only positive exponents. $$ \frac{36 a^{8} b^{2}}{a b} \cdot \frac{a b^{2}}{6} $$

Step-by-Step Solution

Answer

The simplified expression is \(6 a^8 b^3\).

Step 1: Apply the Properties of Exponents

To simplify, use the properties of exponents, namely \( a^m / a^n = a^{m-n} \) and \( (a^m) * (a^n) = a^{m+n} \). Apply these rules to the expression: \( \frac{36 a^{8} b^{2}}{a b} * \frac{a b^{2}}{6} = \frac{36 a^{8-1} b^{2-1}}{6} * \frac{a^1 b^{2}}{6} = 6 a^{8-1} b^{2-1} * a^1 b^{2}\)

Step 2: Simplify Fractions

Simplify the fractions in the expression and combine terms: \( 6 a^7 b^1 * a^1 b^2 = 6 a^{7+1} b^{1+2} = 6 a^8 b^3\)

Step 3: Present the Final Answer

The simplified expression is now \(6 a^8 b^3\). All terms have positive exponents, as required.

Key Concepts

Properties of Exponents

When simplifying algebraic expressions, understanding the properties of exponents is essential. Exponents, also known as powers, are a shorthand way to represent repeated multiplication of the same number. Here's a breakdown of the key properties you'll often use:

• Product of Powers: When multiplying two expressions with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• Quotient of Powers: When dividing two expressions with the same base, you subtract the exponents: \( a^m / a^n = a^{m-n} \).
• Power of a Power: When raising an expression to another exponent, you multiply the exponents: \( (a^m)^n = a^{mn} \).
• Zero Exponent: Any number raised to the power of zero is one: \( a^0 = 1 \), assuming \( a eq 0 \).
• Negative Exponent: A number raised to a negative exponent is the reciprocal of the number raised to the positive exponent: \( a^{-m} = 1/a^m \), for \( a eq 0 \).

Using these properties effectively allows you to simplify expressions efficiently and correctly. For the given exercise, we specifically used the product and quotient of powers rules to combine and reduce the terms.

Positive Exponents

In algebra, it's often preferred to represent expressions using positive exponents. Positive exponents indicate the standard form of repeated multiplication. For example, \( a^3 \) means \( a \times a \times a \). When working with positive exponents, the arithmetic becomes more straightforward, and the expressions are easier to handle.

Expressions with negative exponents can be rewritten with positive exponents by taking the reciprocal of the base. For instance, \( a^{-2} = \frac{1}{a^2} \). In the exercise, we ensured that all exponents remained positive after simplifying. This not only meets the requirements of the exercise but also keeps the expression in a form that is widely accepted and easily understood.

Algebraic Fractions

Algebraic fractions are simply fractions that contain algebraic expressions in the numerator, denominator, or both. Simplifying algebraic fractions involves reducing the fractions to their simplest form, which can include factoring, canceling common factors, and applying the properties of exponents, as needed.

In the given exercise, simplification was done step by step. We first reduced the coefficients (the numerical parts of the terms) by dividing them like ordinary numbers. For the variables, we used the properties of exponents to reduce them. The final expression is an example of an algebraic fraction that has been simplified completely. No factors common to both the numerator and denominator remain, and all variables have positive exponents. When simplifying algebraic fractions, remember to always look for common factors and to apply the properties of exponents carefully to obtain the most simplified form.