Problem 40 - Algebra and Trigonometry

Question

Simplify the expression and eliminate any negative exponent(s). $$ \left(2 s^{3} t^{-1}\right)\left(\frac{1}{4} s^{6}\right)\left(16 t^{4}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(8s^9t^3\).

Step 1: Distribute the expression

Rewrite the given expression by distributing the terms inside each set of parentheses: \[ (2s^3t^{-1})(\frac{1}{4}s^6)(16t^4) \]

Step 2: Simplify each coefficient

Combine the coefficients (numerical values) in the expression:- For coefficients: \(2 \times \frac{1}{4} \times 16 = 2 \times 4 = 8 \)Thus, \(2 \times \frac{1}{4} \times 16\) simplifies to \(8\).

Step 3: Simplify the powers of s

Combine the powers of \(s\) using the property \(a^m \times a^n = a^{m+n}\):- For \(s\): \(s^3 \times s^6 = s^{3+6} = s^9\).

Step 4: Simplify the powers of t

Combine the powers of \(t\) using the property \(a^m \times a^n = a^{m+n}\). Remember that \(t^{-1}\) means the reciprocal, i.e., \(1/t\):- For \(t\): \(t^{-1} \times t^4 = t^{-1+4} = t^3\).

Step 5: Eliminate any negative exponents

Ensure there are no negative exponents in the final expression. We already handled \(t^{-1}\) by adding the exponents, so:Thus, the expression with no negative exponents is:\[ 8s^9t^3 \]

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical expressions that consist of variables, coefficients, and exponents. These expressions can include operations such as addition, subtraction, multiplication, and raising to a power.

Polynomials are very flexible as they can model a wide range of situations. For instance, in our original exercise, we dealt with a polynomial expression that included variables like \(s\) and \(t\), each raised to various powers.

The key components of a polynomial expression are:

• **Terms:** Each individual part of a polynomial, separated by plus or minus signs. For example, in \(8s^9t^3\), each variable part like \(s^9\) and \(t^3\) represents terms.
• **Coefficients:** Numbers that multiply the variable parts. In \(8s^9t^3\), the number 8 is the coefficient.
• **Exponents:** Numbers that indicate how many times a variable is multiplied by itself. For \(s^9\), the 9 is an exponent.

A polynomial expression can be simplified by performing operations like those in the original exercise, which helps in making calculations easier and more efficient.

Exponent Rules

Exponent rules are fundamental guidelines that help in simplifying expressions where variables or numbers are raised to powers (exponents). They are essential for working with polynomial expressions, especially when you have to multiply or divide terms with exponents.

Here are some of the basic rules:

• **Product of Powers Rule:** When multiplying two powers with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\). This was applied in our exercise to simplify each variable, such as \(s^3 \times s^6 = s^9\).
• **Quotient of Powers Rule:** When dividing two powers with the same base, you subtract the exponents: \(a^m / a^n = a^{m-n}\).
• **Power of a Power Rule:** When raising a power to another power, you multiply the exponents: \((a^m)^n = a^{m \times n}\).

Understanding and applying these rules are crucial for simplifying complex expressions, especially because they allow you to rewrite expressions in a simpler form like the culmination in the original final expression \(8s^9t^3\).

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent. For instance, a negative exponent such as \(a^{-n}\) can be rewritten as \(\frac{1}{a^n}\). This is important when simplifying expressions, as it helps eliminate negative exponents by rearranging the expression.

Here's how negative exponents work:

• **Converting Negative to Positive:** Replace the negative exponent by taking the reciprocal: \(t^{-1} = \frac{1}{t}\). This means you move the base with a negative exponent to the other part of the fraction—numerator to denominator or vice versa.
• **Simplifying Expressions:** In the original exercise, \(t^{-1} \times t^4\) becomes \(t^{4-1}\), or \(t^3\), effectively removing the negative exponent by combining terms.
• **Importance in Algebraic Simplification:** Removing negative exponents is crucial for expressing final results neatly and avoiding confusion in calculations.

Negative exponents can seem tricky at first, but with practice, they can be managed efficiently to make expressions simpler.