Problem 138 - Elementary Algebra

Question

How many \(4 a^{2} x^{3}\) 's are there in \(-16 a^{4} x^{5}\) ?

Step-by-Step Solution

Answer

Answer: \(-4a^2x^2\)

Step 1: Divide coefficients

First, we will divide the coefficients of the given expressions. Divide \(-16\) by \(4\) to get \(-4\).

Step 2: Divide variables with respect to exponent

Next, we want to divide the variables separately. Starting with 'a', divide \(a^4\) by \(a^2\). Using the rule \(\frac{a^n}{a^m} = a^{n-m}\), we get \(a^{4-2} = a^2\). Similarly, divide \(x^5\) by \(x^3\). Following the same rule, we get \(x^{5-3} = x^2\).

Step 3: Combine the results

Now, combine the results from Step 1 and Step 2. We get \((-4)(a^2)(x^2)\) which simplifies to \(-4a^2x^2\). So, there are \(-4a^2x^2\) times \(4a^2x^3\)'s in \(-16a^4x^5\).

Key Concepts

Exponents

Exponents are a way to represent repeated multiplication of a number. When you see something like \(a^4\), it means you multiply \(a\) by itself four times (i.e., \(a \times a \times a \times a\)). Exponents follow specific rules that help us simplify algebraic expressions. One important rule is the division of exponents used in expressions like \(\frac{a^n}{a^m} = a^{n-m}\). This rule helps us in simplifying expressions by subtracting the exponent of the divisor from the exponent of the dividend, which was crucial in solving the given exercise.

Dividing Coefficients

Coefficients are the numerical parts of terms in an algebraic expression. In our exercise, the terms \(4a^2x^3\) and \(-16a^4x^5\) have coefficients 4 and -16, respectively. Dividing coefficients involves simply dividing the numbers. For instance, when dividing \(-16\) by \(4\), you perform regular division to get \(-4\). This easy step is essential as it separates the numerical part from the variable part, allowing us to focus separately on the variables using exponent rules.

Combining Like Terms

Combining like terms is a fundamental aspect of simplifying algebraic expressions. Like terms have the same variable parts, meaning both the variables and their exponents must match. For instance, \(3x^2\) and \(5x^2\) are like terms because they both have \(x^2\). In the exercise solution, we first simplified the expression by dividing each component and combined the resulting terms to get \(-4a^2x^2\). It's important to remember that only like terms can be combined to get a simpler form.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and exponents connected by operations like addition, subtraction, multiplication, and division. The exercise involves dividing two algebraic expressions \(4a^2x^3\) from \(-16a^4x^5\). Such expressions can be simplified using mathematical operations and rules like the ones for exponents and coefficients. Algebraic expressions are essential parts of algebra and understanding how to handle them, including simplifying through division, is vital for solving many math problems.