Problem 125 - Algebra and Trigonometry

Question

Why is \(\left(-3 x^{2}\right)\left(2 x^{-5}\right)\) not simplified? What must be done to esimplify the expression?

Step-by-Step Solution

Answer

The simplified expression for \(\left(-3 x^{2}\right)\left(2 x^{-5}\right)\) is \(-6/x^{3}\).

Step 1: Recognize the rules for multiplying exponents

When multiplying expressions with the same base (in this case, the base is x), we add the exponents. Check if the terms can be multiplied together using this rule.

Step 2: Multiply the coefficients and add the exponents

Multiply the coefficients together (-3 * 2 = -6) and add the exponents together (2 - 5 = -3). So, the expression should be simplified to \(-6 x^{-3}\).

Step 3: Rewrite the expression into standard form

Exponents in mathematics are usually given in non-negative form. Therefore, we can rewrite \(x^{-3}\) as \(1/x^{3}\). So, the final simplified expression is \(-6/ x^{3}\).

Key Concepts

Multiplying Exponents

Understanding the process of multiplying exponents is crucial for simplifying algebraic expressions effectively. When two algebraic terms with the same base are multiplied, their exponents should be added together. For example, with the expression \(x^a \times x^b\), the simplified form would be \(x^{a+b}\).

Let's look at an example to make this clearer. Considering \(x^2\times x^5\), we simply add the exponents because the base, \(x\), is the same for both terms. Thus, we get \(x^{2+5} = x^7\). It's important to understand that this rule only applies when the bases are identical; otherwise, the expression cannot be simplified in this way.

This principle was applied in the step by step solution of the original exercise where we had \(x^2 \times x^{-5}\) resulting in a new exponent by adding \(2 + (-5)\) to get \(x^{-3}\).

Negative Exponents

Dealing with negative exponents is a common challenge. In essence, a negative exponent indicates that the base is on the 'wrong side of the fraction line.' In mathematical terms, a negative exponent translates to its positive counterpart in the denominator. For instance, \(x^{-a}\) equals \(1/x^a\).

To clarify, take \(x^{-3}\). This would be expressed as \(1/x^3\), meaning that \(x\) is raised to the third power in the denominator, not the numerator. The negative exponent essentially 'flips' the position of the base from the top to the bottom of a fraction.

In our original problem, after combining the coefficients and adding up the exponents, we arrived at \(x^{-3}\), which is not the standard form for expressing exponents. To convert it into a positive one, we rewrote \(x^{-3}\) as \(1/x^3\), as shown in step 3 of the solution.

Coefficients in Algebra

The term coefficient in algebra refers to the numerical part of an algebraic term which is multiplied by a variable or a product of variables. For instance, in the term \(5x^2\), \(5\) is the coefficient. When simplifying algebraic expressions, coefficients are multiplied together while the variables are treated separately according to the rules of exponents.

When we look at the exercise provided, \(\left(-3 x^{2}\right)\left(2 x^{-5}\right)\), we address the coefficients \(\left(-3\right)\) and \(\left(2\right)\) first. They are multiplied to give us \(\left(-3 \times 2 = -6\right)\), which is the new coefficient for the combined variable term.

In algebra, it's essential to keep clear the distinction between coefficients and variables, as mistaking one for the other can lead to incorrect simplification of expressions. The correct treatment of coefficients, as exhibited in the provided solution, is a fundamental aspect of algebra that allows us to simplify the expression to its standard form.