Problem 43 - Elementary Algebra

Question

Write the expressions for the following problems using only positive exponents. $$ \frac{16 a^{5} b^{-2}}{-2 a^{3} b^{-5}} $$

Step-by-Step Solution

Answer

Question: Rewrite the given expression using only positive exponents: $\frac{16 a^{5} b^{-2}}{-2 a^{3} b^{-5}}$ Answer: $-8 a^{2} b^{3}$

Step 1: Rewrite the terms with negative exponents

Move the terms with negative exponents to the opposite part of the fraction and change their exponents to positive: $$ \frac{16 a^{5} b^{-2}}{-2 a^{3} b^{-5}} = \frac{16 a^{5}}{-2 a^{3}} \cdot \frac{b^{5}}{b^{2}} $$

Step 2: Simplify the factors in the expression

Simplify the numerical factors and use the exponent rule \(a^{m} \div a^{n} = a^{m-n}\) for the exponential terms: $$ \frac{16}{-2} = -8 $$ $$ a^{5} \div a^{3} = a^{5-3} = a^{2} $$ $$ b^{5} \div b^{2} = b^{5-2} = b^{3} $$

Step 3: Combine the simplified factors

Combine the simplified numerical and exponential factors into a single expression: $$ -8 a^{2} b^{3} $$ So, the given expression with only positive exponents is: $$ \frac{16 a^{5} b^{-2}}{-2 a^{3} b^{-5}} = -8 a^{2} b^{3} $$

Key Concepts

Simplifying Algebraic Expressions

In algebra, simplifying expressions is a fundamental skill that enables you to reduce a complex equation into its simplest form. This often involves combining like terms, canceling out terms, and applying mathematical operations in the correct order.

Like terms are those that have the same variables raised to the same power. For example, in the expression \(2x + 3x\), we can combine the like terms to get \(5x\). Canceling out terms usually occurs in fractions where a term in the numerator is the same as one in the denominator, allowing us to reduce the fraction.

It's also vital to perform operations following the order of precedence. Operations inside parentheses come first, followed by exponents, multiplication and division, and finally addition and subtraction. Through these steps of combining, canceling, and adhering to operation order, we can simplify an algebraic expression efficiently.

Negative Exponents

Negative exponents in algebra indicate that the base of the exponent should be reciprocated or 'flipped'. In simpler terms, a number with a negative exponent, such as \(a^{-n}\), can be rewritten as \(\frac{1}{a^n}\).

This 'flipping' feature lets us turn negative exponents into positive ones, making the expression easier to work with. For instance, if we have a fraction with a negative exponent in the denominator, like \(\frac{1}{a^{-2}}\), this can be transformed into \(a^2\) in the numerator.

Understanding how to work with negative exponents is crucial since it allows us to maintain expressions in a form that does not involve negative powers—often a requirement in final answers for mathematical problems. Furthermore, keeping all exponents positive can simplify the calculation process and help avoid mistakes.

Exponent Rules

Exponent rules, also known as laws of exponents, are guidelines for performing operations on powers. When multiplying numbers or variables with the same base, you add the exponents: \(a^m \cdot a^n = a^{m+n}\). Conversely, when dividing, you subtract the exponents as seen in \(\frac{a^m}{a^n} = a^{m-n}\).

For raising a power to another power, you multiply the exponents: \(\left(a^m\right)^n = a^{m\cdot n}\). Another rule is for zero powers: any base (except for zero) to the power of zero is one, meaning \(a^0 = 1\), as long as \(a \eq 0\).

There's also a rule for one as the power, which is simple yet essential: any base to the power of one equals the base itself, so \(a^1 = a\). Remembering and applying these exponent rules will help you simplify expressions and solve equations more efficiently and accurately. Utilizing these rules in the given exercise transformed the complex fraction into an expression with positive exponents only.