Problem 59 - Algebra 1

Question

Simplify the expression. (Review \(8.3 \text { for } 11.7)\) $$\frac{42 x^{4} y^{3}}{6 x^{3} y^{9}}$$

Step-by-Step Solution

Answer

The simplified expression is \(7x/y^{6}\)

Step 1: Simplify Numerical Coefficients

The numerical coefficients are 42 and 6. Divide 42 by 6 to simplify the coefficients. It simplifies to 7.

Step 2: Simplify Variables with Exponents

For variables with exponents, subtract the exponent of the denominator from the exponent of the numerator. In case of x's, subtract the exponent 3 (denominator) from 4 (numerator) which results in 1. For y's subtract the exponent 9 (denominator) from 3 (numerator) which results in -6.

Step 3: Rewrite the expression

With the simplified coefficient 7, \(x^{1}\) or simply \(x\) in the numerator, and \(y^{-6}\) or \(1/y^{6}\) in the denominator, rewrite the expression to simplify.

Step 4: Simplified Expression

Finally, by moving \(y^{6}\) from the denominator to the numerator (to get rid of the negative exponent), one can obtain the simplified expression.

Key Concepts

Numerical Coefficients

When simplifying algebraic expressions, the process often starts with the numerical coefficients. These are the numbers in front of the variables. In our example, the expression \(\frac{42 x^{4} y^{3}}{6 x^{3} y^{9}}\) has numerical coefficients of 42 and 6. To simplify, you divide the larger coefficient by the smaller one, leading to \(\frac{42}{6} = 7\). This step effectively reduces the complexity of the problem by removing larger numbers earlier on, allowing a clearer view of the variables and their exponents.

Understanding numerical coefficients is essential because they have a direct impact on the value of the expression. Remember, always seek the greatest common divisor when simplifying these coefficients. This can make subsequent steps much more manageable and help prevent errors in simplification.

Variables with Exponents

Variables with exponents can sometimes be intimidating, but with the correct approach, they can be simplified effectively. For the given problem, we have \( x^{4} \) and \( x^{3} \) for the variable \(x\), and \( y^{3} \) and \( y^{9} \) for the variable \(y\). When simplifying, subtract the exponent in the denominator from the exponent in the numerator for each variable. This is based on the laws of exponents, particularly \( x^{a} / x^{b} = x^{a-b} \).

Applying this to our variables, \( x^{4-3} \) equals \( x^{1} \) or simply \( x \), and for \( y \) we calculate \( y^{3-9} \) which simplifies to \( y^{-6} \). It’s worth noting when an exponent is negative, it means that the variable factor is in the wrong part of the fraction—what was in the numerator should be in the denominator and vice-versa.

Simplifying Fractions in Algebra

Simplifying fractions in algebra is closely related to simplifying numerical fractions. However, instead of numbers, we deal with expressions containing variables and exponents. After you've dealt with the numerical coefficients and variables separately, the next step is to rewrite the simplified expression.

In our example, after simplifying, we get \(7x / y^6\). Understanding that a negative exponent indicates the reciprocal of the base raised to the positive of that exponent is key. Thus, \(y^{-6}\) is equal to \(1/y^6\). When we apply this knowledge, we can rewrite the expression to place the variable with the negative exponent back in the correct position, giving us our final simplified expression. Always remember, the objective is to express our solution with positive exponents and in the most reduced form possible.