Problem 32 - Elementary Algebra

Question

For the following problems, find the products. $$ (6 t-7 s)^{2} $$

Step-by-Step Solution

Answer

Question: Find the product of the given expression: \((6t - 7s)^2\). Answer: The product of the given expression is \(36t^2 - 84ts + 49s^2\).

Step 1: Recall the binomial square formula

Recall that when a binomial expression, such as \((a + b)^2\), is squared, it can be expanded using the formula: \((a + b)^2 = a^2 + 2ab + b^2.\) In this case, \(a = 6t\) and \(b = -7s.\)

Step 2: Substitute the values in the formula

Now, substitute these values into the formula: \((6t - 7s)^2 = (6t)^2 + 2(6t)(-7s) + (-7s)^2.\)

Step 3: Simplify each term

Next, we need to simplify each term in the expression. To do this, we apply the rules of exponents and multiplication: \((6t)^2 = (6^2)(t^2) = 36t^2,\) \(2(6t)(-7s) = 2\times 6\times (-7)ts = -84ts,\) \((-7s)^2 = (-7)^2(s^2) = 49s^2.\)

Step 4: Combine the simplified terms

Lastly, combine the simplified terms to obtain the final expression: \((6t - 7s)^2 = 36t^2 - 84ts + 49s^2.\) The product of the given expression is \(36t^2 - 84ts + 49s^2.\)

Key Concepts

Algebra

Algebra is a fundamental branch of mathematics that involves the study of symbols and the rules for manipulating these symbols. It's much like a language where letters and symbols represent numbers and unique mathematical operations.
For expressions like \(6t - 7s\)^2, understanding algebra allows us to expand or simplify them effectively.Let's break down a few key aspects of Algebra:

• **Variables and Constants**: Variables represent numbers we don't know yet or any number in a range, denoted by symbols like \(t\) and \(s\). Constants, on the other hand, are specific fixed numbers such as \(-7\) or \ (6)\.
• **Operations**: Algebra uses basic arithmetic operations like addition, subtraction, multiplication, division, and allows more complicated operations through exponents or roots.
• **Expressions and Equations**: An algebraic expression is a combination of numbers, variables, and operations, like \(6t - 7s\). In contrast, equations are statements of equality, meaning two expressions are equal, like \(6t - 7s = x\).

Algebra uses simplification and factoring to solve problems and perform calculations efficiently, as demonstrated in binomial expansions.

Polynomial expressions

Polynomial expressions are an essential component of algebra, consisting of variables raised to whole number exponents and coefficients. Each polynomial is a sum of these terms. For example, \(36t^2 - 84ts + 49s^2\) is a polynomial expression.Here are a few things to note about polynomial expressions:

• **Terms**: Each distinct piece in a polynomial, separated by a "+" or a "-", is a term. In \(36t^2 - 84ts + 49s^2\), the terms are \(36t^2, -84ts\), and \(49s^2\).
• **Degree**: This is determined by the highest exponent of the variable. Here, the degree is 2 because each term has variables raised to a power of 2.
• **Coefficients**: These are numbers in front of the variables. In \(36t^2\), 36 is the coefficient.

Polynomial expressions can be added, subtracted, multiplied, and sometimes divided, allowing for various manipulations and solutions to algebraic problems.

Exponents

Exponents denote repeated multiplication of a number by itself and are a critical component of algebraic expressions. In \(6t - 7s\)^2, the exponent 2 tells us to multiply the binomial \(6t - 7s\) by itself.Understanding exponents involves:

• **Base and Exponent**: In expressions like \(t^2\), \(t\) is the base, while 2 is the exponent, indicating \(t\) is multiplied by itself once.
• **Properties of Exponents**: Key properties include \(a^m \times a^n = a^{m+n}\) and \(\frac{a^m}{a^n} = a^{m-n}\). Learning these makes it easier to work with polynomials.
• **Special Cases**: Be mindful of cases like \(a^0 = 1\) for any non-zero \(a\) and \(a^{-n} = \frac{1}{a^n}\).

Exponents help simplify expressions and perform calculations in sequences and series, forming an essential tool in algebra and beyond.