Problem 71 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ 3 a^{5} b^{-7}\left(a^{2}+4\right)^{-3} 6 a^{-4} b\left(a^{2}+4\right)^{-1}\left(a^{2}+4\right) $$

Step-by-Step Solution

Answer

Question: Rewrite the expression with positive exponents: \(\displaystyle \frac{3a^{5}}{b^{7}(a^{2} + 4)^{3}} \cdot 6\frac{b}{a^{4}(a^{2}+4)}(a^{2}+4)\) Answer: \(\displaystyle \frac{18a}{b^{6}(a^{2}+4)}\)

Step 1: Rewrite using positive exponents

In order to rewrite the negative exponents in the given expression, we can recall the rule \(\displaystyle a^{-m} =\dfrac{1}{a^{m}} \). Using this rule, we can rewrite our expression as: $$ \frac{3 a^{5}}{b^{7}(a^{2}+4)^{3}} \cdot 6 \frac{b}{a^{4}(a^{2}+4)^{1}} (a^{2}+4) $$

Step 2: Multiply terms

Now, we can multiply the terms together. We'll group the numbers, the variables \(a\) and \(b\), and the polynomial \((a^{2} + 4)\): $$ \frac{3 \cdot 6}{1} \cdot \frac{a^{5}}{a^{4}} \cdot \frac{b}{b^{7}} \cdot \frac{(a^{2}+4)}{(a^{2}+4)^3} \cdot (a^{2}+4) $$

Step 3: Simplify numbers and variables

Now, let's simplify the numbers and variables in the expression: $$ \frac{18}{1} \cdot \frac{a^{5-4}}{1} \cdot \frac{1}{b^{7-1}} \cdot \frac{(a^{2}+4)^{2-3}}{1} $$ Applying the simplified values, we get: $$ 18 \cdot a^{1} \cdot \frac{1}{b^{6}} \cdot (a^{2}+4)^{-1} $$

Step 4: Rewrite using only positive exponents

Now, we'll rewrite the expression using only positive exponents. For the \((a^{2}+4)^{-1}\) term, we can recall the rule \(\displaystyle a^{-m} =\dfrac{1}{a^{m}} \). Using this rule, we get: $$ 18 a \cdot \frac{1}{b^{6} (a^{2}+4)} $$

Step 5: Final simplified expression

Finally, we express the simplified expression using only positive exponents: $$ \frac{18a}{b^{6} (a^{2}+4)} $$ So, the given expression simplified and written with only positive exponents is: \(\displaystyle \frac{18a}{b^{6}\left(a^{2}+4\right)}\).

Key Concepts

Exponent Rules

Exponents are a way of representing repeated multiplication of a number by itself. In algebra, applying exponent rules can simplify expressions and make them easier to handle. One important rule is that when you have a negative exponent, such as \(a^{-m}\), it becomes \(\frac{1}{a^m}\). This indicates that the base \(a\) is on the denominator side of the fraction.
When you multiply terms with the same base, you add their exponents together, following the rule \(a^m \cdot a^n = a^{m+n}\). This rule helps in condensing expressions with multiple exponents. Similarly, when you divide terms with the same base, subtract their exponents: \(\frac{a^m}{a^n} = a^{m-n}\). These rules are crucial for rewriting expressions with positive exponents and simplifying them effectively.
Using these rules, you can transform complicated algebraic expressions into simpler forms, making evaluations and calculations much easier.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operators. Variables are symbols that stand for unknown or arbitrary numbers, like \(a\) and \(b\) in our exercise. Algebraic expressions can range from simple terms such as \(3a\) to more complex forms like \(3a^5b^{-7}(a^2+4)^{-3}6a^{-4}b(a^2+4)^{-1}(a^2+4)\).
Understanding the structure of expressions is crucial for manipulating and simplifying them. Each part can contribute differently: coefficients (like 3 and 6) scale the terms, exponents indicate repeated multiplication, and operators (such as '+' and '-') determine how components are combined.
By recognizing these elements, you can apply the correct mathematical rules and strategies to simplify expressions. Whether it's distributing multiplication across terms or dealing with exponents using algebraic rules, breaking down the expression into manageable parts is key to understanding.

Simplifying Expressions

Simplifying expressions involves using algebraic rules to transform them into their simplest form. The process often starts by identifying and applying exponent rules to handle any negative exponents. For instance, converting \(b^{-7}\) into \(\frac{1}{b^7}\) helps in rewriting the expression in a form that's easier to manipulate.
Once all terms are expressed with positive exponents, combine like terms. This includes adding or subtracting exponents as necessary when dealing with similar bases. Multiplying constants, such as in \(3 \cdot 6\), gives you a product of 18, which simplifies the numerical part of the expression.
After combining and simplifying each component, the expression should reflect the simplest version with positive exponents only. The process can be challenging at times, especially when dealing with complex terms, but it becomes manageable by methodically applying math rules and breaking down the expression into smaller, more comprehensible parts. The final expression is easier to interpret and use in further calculations, as seen with \(\frac{18a}{b^6(a^2+4)}\).