Problem 7 - Elementary Algebra

Question

For Problems \(1-30\), multiply using the properties of exponents to help with the manipulation. $$ \left(-2 x^{2} y\right)(-7 x) $$

Step-by-Step Solution

Answer

The product is \(14x^3y\).

Step 1: Identify the Factors

The expression given to us is \((-2x^2 y)(-7x)\). We'll separate this into individual parts: \(-2x^2y\) and \(-7x\).

Step 2: Combine the Numerical Coefficients

Multiply the numerical coefficients of each part. Here, we will multiply \(-2\) by \(-7\), which gives us: \[ (-2)\times (-7) = 14 \]

Step 3: Combine Like Variables

Combine the variables with like bases by adding their exponents. For \(x\), we have \(x^2\) and \(x^1\), so: \[x^{2+1} = x^3 \] There is no other \(y\) term to combine, so it stays as it is.

Step 4: Multiply Everything Together

Now, multiply the results we've obtained: \[ 14 \times x^3 \times y = 14x^3y \] This combines the coefficients and the variables into one expression.

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations (such as addition, subtraction, multiplication, and division). These expressions represent mathematical relationships and can be manipulated using various algebraic rules. For instance, in the expression \((-2x^2 y)(-7x)\), we can observe:

• The presence of numbers, which are the coefficients \(-2\) and \(-7\).
• Variables, represented here by \(x\) and \(y\).
• Operations, mainly multiplication in this case.

Understanding algebraic expressions is essential because they form the foundation for solving equations and performing more complex operations in algebra. In particular, recognizing and combining like terms helps simplify expressions, making them easier to work with.

Multiplying Variables

Multiplying variables involves applying the rules of exponents. When variables are multiplied together, the exponents of like bases are added. This is because multiplying variables is essentially repeated addition of their exponents. For instance, when you see \(x^2\) and \(x^1\), you are combining these by adding the exponents: \[x^{2+1} = x^3\].A few key points to remember about multiplying variables:

• If the bases are the same, add the exponents. If the bases are different, keep each base's variable separate in your final expression.
• This principle applies regardless of the number of variables. Hence, \(y^1\) remains \(y\), as there are no other \(y\) terms to combine.

This method of simplification is foundational in algebra since it helps condense and clarify expressions for further solving or graphing.

Exponents

Exponents are a shorthand way to indicate how many times a number (called the base) is multiplied by itself. In the expression \(x^2\), for example, the number 2 is the exponent and tells us that \(x\) is multiplied by itself once (i.e., \(x \times x\)).Here are some essential properties of exponents that are useful in manipulating algebraic expressions:

• Product of Powers: \(a^m \times a^n = a^{m+n}\) — This rule was used in the provided solution for combining \(x^2\) and \(x^1\).
• Power of a Power: \((a^m)^n = a^{m\cdot n}\) — Not directly applied here but crucial in more complex problems.
• Zero Exponent: Any base raised to the zero power is 1 (\(a^0 = 1\)), which can simplify other expressions.

In our example, understanding the product of powers property allows us to quickly and accurately compile the components of an algebraic expression, making exponents a powerful tool in algebra.