Problem 50 - Algebra and Trigonometry

Question

\(47-72\) . Simplify the expression, and eliminate any negative exponent(s). $$ \left(2 s^{3} t^{-2}\right)\left(\frac{1}{4} s^{7} t\right)\left(16 t^{4}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(8 s^{10} t^{3}\).

Step 1: Expression Setup

We start with the given expression: \( \left(2 s^{3} t^{-2}\right)\left(\frac{1}{4} s^{7} t\right)\left(16 t^{4}\right) \). Our goal is to simplify this expression while eliminating any negative exponents.

Step 2: Multiply Constants Coefficients

First, let's multiply the numerical coefficients of each term: \(2\), \(\frac{1}{4}\), and \(16\). Specifically, \(2 \times \frac{1}{4} \times 16 = 8\). So, the numerical part of the expression becomes \(8\).

Step 3: Simplify the \(s\) Terms

Now, we'll combine the powers of \(s\) by adding their exponents:- \(s^3\) from first term- \(s^7\) from the second termAdding the exponents: \(3 + 7 = 10\), thus we have \(s^{10}\).

Step 4: Simplify the \(t\) Terms

Next, we need to combine the powers of \(t\) by adding their exponents:- \(t^{-2}\) from the first term- \(t^1\) from the second term (note: \(t\) is equivalent to \(t^1\))- \(t^4\) from the third termAdding these exponents: \(-2 + 1 + 4 = 3\), thus we have \(t^{3}\).

Step 5: Write Final Simplified Expression

Combine the results of the previous steps to write the simplified expression:\[8 s^{10} t^{3}\]

Key Concepts

Simplifying Expressions

When simplifying expressions, our goal is to rewrite the expression in a simpler, more compact form without changing its value. In algebra, simplification often involves combining like terms and performing multiplications or divisions where possible. For this particular exercise, we're dealing with an expression that involves products of coefficients and variables raised to powers.

The process begins by examining the expression and deciding how to combine terms effectively. We start by identifying like terms, which are terms that involve the same variable raised to any power. In the given expression, for example, we see terms with the variables \(s\) and \(t\), and we simplify these by adding their exponents whenever we multiply them together. This method ensures all similar parts of the expression are unified, allowing for simpler and more understandable expressions.

Moreover, when dealing with numerical coefficients, they are multiplied together separately from the variables. The separate multiplication ensures that the process of simplification is straightforward and easily manageable.

Exponents

Exponents are a convenient way to express repeated multiplication of the same factor. In this exercise, exponents are employed to simplify expressions involving variables. For example, \(s^3\) means \(s\) multiplied by itself three times. Likewise, \(t^4\) represents four consecutive multiplications of \(t\). Exponents offer a tidy shorthand notation that makes algebraic expressions easier to work with.

When multiplying terms with the same base (the same variable), you add their exponents to simplify the expression. For instance, when multiplying \(s^3\) by \(s^7\), you add the exponents: \(3 + 7 = 10\), resulting in \(s^{10}\). This rule comes in handy for swiftly reducing the number of terms or operations you must deal with in complex expressions.

Remember, exponentiation is a powerful tool in algebra. Understanding how to manipulate these exponents, especially how to deal with them during multiplication and division, can dramatically streamline your mathematical work.

Negative Exponents

Negative exponents pose a little more complexity because they signify the reciprocal of a number. For example, \(t^{-2}\) can be rewritten as \(\frac{1}{t^2}\). This tells us that instead of multiplying by \(t^2\), we are dividing by it. Negative exponents are essential for expressing fractions without using division symbols directly in multiplication.

Changing negative exponents into positive ones is a common step in simplifying expressions. In our exercise, the term \(t^{-2}\) becomes particularly relevant, as we aim to express the entire result using non-negative exponents. By moving \(t^{-2}\) from the numerator to the denominator of a fraction (or simply eliminating the negative sign through combination with other \(t\) terms), we can often simplify the expression significantly.

Working with negative exponents effectively can aid in solving equations and is helpful in scenarios where upside-down relationships between variables or coefficients need clear expression.