Problem 72 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

Question: Rewrite the given expression with positive exponents: $$-4 a^{3} b^{-5}(2 a^{2} b^{7} c^{-2})$$. Answer: The given expression can be rewritten with positive exponents as follows: $$\frac{-8 a^{5}}{b^{5}} - \frac{4 a^{3} b^{2}}{c^{2}}$$.

Step 1: Rewrite the given expression

We will rewrite the given expression as follows: $$ -4 a^{3} b^{-5}\left(2 a^{2} b^{7} c^{-2}\right) $$

Step 2: Apply the distributive property

In order to do so, we need to distribute the terms inside the parenthesis with the terms outside the parenthesis. This will help us combine the terms with same variable: $$ = -4 a^{3} b^{-5}(2 a^{2}) + (-4 a^{3} b^{-5})(b^{7} c^{-2}) $$

Step 3: Simplify and apply the rules of exponents

Now, we will simplify and apply the rules of exponents to rewrite the expression with positive exponents: $$ = -8 a^{3+2} b^{-5} + (-4 a^{3}) b^{-5+7} c^{-2} $$ $$ = -8 a^{5} b^{-5} - 4 a^{3} b^{2} c^{-2} $$

Step 4: Rewrite the expression with only positive exponents

In this step, we will use the rule (a^{-n} = 1 / a^{n}) to rewrite the negative exponents as positive exponents: $$ = \frac{-8 a^{5}}{b^{5}} - \frac{4 a^{3} b^{2}}{c^{2}} $$ We have now rewritten the entire expression using only positive exponents.

Key Concepts

Distributive Property

The distributive property is a useful algebraic tool that helps us simplify expressions. When we talk about the distributive property in algebra, it generally means distributing multiplication over addition or subtraction. In mathematical terms, for any numbers or variables, the rule is:

• If you have \( a(b + c) \), it will become \( ab + ac \).
• This applies similarly to subtraction, \( a(b - c) = ab - ac \).

When dealing with expressions that include exponents, like \(-4a^3b^{-5}(2a^2b^7c^{-2})\), we apply this property by multiplying each term inside the parentheses by the term outside, allowing us to manage and combine the like terms effectively. It's a key step for simplifying complex algebraic expressions and plays a crucial role in solving equations.

Negative Exponents

Exponents are a way of expressing repeated multiplication of a number by itself. When the exponent is positive, it's straightforward: \( a^3 = a \times a \times a \). However, negative exponents can seem confusing at first, but they follow a simple rule. A negative exponent indicates a reciprocal.

• For any nonzero number \( a \), \( a^{-n} = \frac{1}{a^n} \).
• Negative exponents mean "take the reciprocal and make the exponent positive."

For instance, \( b^{-5} = \frac{1}{b^5} \). When simplifying algebraic expressions, converting negative exponents to positive exponents by using their reciprocal form makes further mathematical work much easier.

Positive Exponents

Once we understand how to handle negative exponents using their reciprocal, it becomes easier to work with positive exponents, which are more straightforward and common in expressions. Positive exponents indicate the number of times you multiply the base by itself. When managing expressions, such as the converted form in the original problem, we see expressions like \( a^5 \) or \( b^2 \), which imply repeated multiplication in a straightforward manner:

• \( a^5 = a \times a \times a \times a \times a \)
• \( b^2 = b \times b \)

Working with positive exponents simplifies the process of evaluation and further algebraic manipulations, because it is straightforward to calculate the resulting products.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and exponents combined using mathematical operations like addition, subtraction, multiplication, and division. Understanding the structure and properties of these expressions is crucial for any algebraic problem-solving.

• An example of an algebraic expression is \(-4a^3b^{-5}(2a^2b^7c^{-2})\).
• They can look complex but are built from simpler components following consistent rules.

The exponents indicate the power to which a variable is raised and can dramatically affect the value of the expression. Handling these expressions often involves simplifying them by applying properties like distributive, commutative, and associative, as well as rules for exponents. With practice, interpreting and manipulating algebraic expressions becomes more intuitive.