Problem 51 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ \frac{\left(6 a^{2} b^{8}\right)^{2}}{\left(3 a b^{5}\right)^{2}} $$

Step-by-Step Solution

Answer

Question: Simplify the following expression: \(\frac{\left(6 a^{2} b^{8}\right)^{2}}{\left(3 a b^{5}\right)^{2}}\) Answer: \(4a^2b^6\)

Step 1: Apply the power rule

To raise the terms inside the parentheses to the power, we need to apply the power rule to each term: Power Rule: \((a^{m})^{n} = a^{mn}\) Apply this rule to both the numerator and the denominator: $$ \frac{\left(6^{2} a^{2\cdot2} b^{8\cdot2}\right)}{\left(3^{2} a^{1\cdot2} b^{5\cdot2}\right)} $$

Step 2: Calculate the powers

Now, we need to calculate the powers of each term: $$ \frac{\left(36 a^{4} b^{16}\right)}{\left(9 a^{2} b^{10}\right)} $$

Step 3: Cancel common terms

Now we will cancel common terms in the numerator and denominator: $$ \frac{36}{9} = 4 $$ $$ \frac{a^{4}}{a^{2}} = a^{4-2} = a^2 $$ $$ \frac{b^{16}}{b^{10}} = b^{16-10} = b^6 $$

Step 4: Rewrite the simplified expression

Finally, we will rewrite the simplified expression: $$ \frac{\left(6 a^{2} b^{8}\right)^{2}}{\left(3 a b^{5}\right)^{2}} = 4a^2b^6 $$ The simplified expression is \(4a^2b^6\).

Key Concepts

Simplifying Expressions

Simplifying expressions is like tidying up a room. Just as you organize items neatly, we must organize mathematical expressions for clarity and ease of use. It involves reducing expressions to their simplest form. This means getting rid of any unnecessary or redundant parts. For instance, simplifying can include combining like terms, using operations and rules like the power rules for exponents.

When working with exponents, the power rule is your best friend. The power rule states: \((a^{m})^{n} = a^{mn}\). This means when you have a power raised to another power, you multiply the exponents. By applying this rule, we reduce complicated exponential expressions into something manageable. Simplifying isn't just limited to reducing powers; it's about finding ways to make your math expressions as concise as possible. This approach is crucial in both solving problems and in conveying solutions clearly.

Algebraic Fractions

Algebraic fractions, much like ordinary fractions, consist of a numerator and a denominator. However, instead of simple numbers, these fractions contain algebraic terms. Working with algebraic fractions involves similar principles as numeric fractions. You simplify them by canceling out common factors in the numerator and the denominator.

Consider the fraction \(\frac{36 a^{4} b^{16}}{9 a^{2} b^{10}}\) from our problem. To simplify it, we divide both the numerator and the denominator by their greatest common factor. We simplify the numeric factors first (\(36/9 = 4\)), and then handle the variables by using the rule \(\frac{a^m}{a^n} = a^{m-n}\). This method helps in reducing algebraic fractions to simpler forms, making the expressions easier to manage and solve.

Remember, simplifying algebraic fractions is a fundamental skill in algebra that helps with solving more complex equations and problems.

Natural Numbers

Natural numbers are simply the set of positive integers starting from 1, and they go on infinitely. In algebra, natural numbers are frequently used as exponents in expressions, representing multiplication of a base number. For example, in \(a^3\), the number "3" is a natural number and tells you to multiply "a" by itself three times.

When simplifying expressions with natural number exponents, you apply specific rules, like the power rule, to systematically solve or simplify the expression. Natural numbers are easy to work with since they don’t include negative numbers or fractions. However, understanding their properties and how they function within expressions is essential. They are fundamental in various mathematical operations and are vital in mastering algebra and other higher-level math topics.

• They help in structuring exponential expressions.
• They simplify computations by reducing large expressions to manageable sums or products.
• They offer a straightforward way to express repeated multiplication.

By grasping the concept of natural numbers and their role in exponents, you gain better control over arithmetic and algebraic manipulations.