Problem 22 - Elementary Algebra

Question

\(-\frac{18 p^{7}}{40 p}\)

Step-by-Step Solution

Answer

-\frac{9}{20} p^{6}

Step 1 - Simplify the coefficients

The expression \(-\frac{18 p^{7}}{40 p}\)\ involves two numbers: -18 and 40. Simplify the fraction \(-\frac{18}{40}\) by finding the greatest common divisor (GCD) of 18 and 40. The GCD is 2, so \(-\frac{18}{40} = -\frac{9}{20}\).

Step 2 - Simplify the exponents

Next, simplify the exponent part \(p^{7}\) divided by \(p\). Using the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\), \(\frac{p^{7}}{p} = p^{7-1} = p^{6}\).

Step 3 - Combine the simplified results

Combine the results from Step 1 and Step 2 to get the simplified expression. Thus, \(-\frac{18 p^{7}}{40 p}\) simplifies to \(-\frac{9}{20} p^{6}\).

Key Concepts

Coefficients

In algebra, coefficients are the numerical part of terms that include variables. They tell us how many times to multiply the variable. For example, in the expression \(-\frac{18 p^{7}}{40 p}\), the numbers -18 and 40 are the coefficients of the terms with the variable \('p'\). To simplify the coefficients, we divide them by their greatest common divisor (GCD). Here, we have -18 and 40. The GCD of 18 and 40 is 2. Thus, \(-\frac{18}{40}\) simplifies to \(-\frac{9}{20}\). Understanding coefficients helps us reduce fractions in algebraic expressions and makes solving problems much simpler. Never forget to simplify coefficients as much as possible before you move on to the variables.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides two or more integers without leaving a remainder. It is useful when simplifying fractions or algebraic expressions. For example, to simplify \(-\frac{18}{40}\), we need to find the GCD of 18 and 40. Let's break it down:

• 18 can be divided by 1, 2, 3, 6, 9, and 18.
• 40 can be divided by 1, 2, 4, 5, 8, 10, 20, and 40.

The largest number common to both sets is 2. Therefore, the GCD is 2.Once we have the GCD, we can simplify the fraction by dividing both the numerator and the denominator by this number. So, \(-\frac{18}{40} = -\frac{9}{20}\). Using the GCD ensures that we simplify fractions completely, making the expressions easier to work with.

Properties of Exponents

Exponents represent repeated multiplication of a base number. For example, \(a^m\) means the base \(a\) is multiplied by itself \(m\) times. When simplifying expressions involving exponents, some properties become very handy.Let's focus on the property \(\frac{a^m}{a^n} = a^{m-n}\). This rule tells us that when you divide powers with the same base, you subtract the exponents. In the expression \(-\frac{18 p^{7}}{40 p}\), we simplify the exponents by applying this property:

• We have \(p^7\) in the numerator and \(p\) in the denominator.
• Subtracted the exponents: \(p^{7-1} = p^6\).

Combining these simplified exponents with the simplified coefficients from the previous steps results in the expression \(-\frac{9}{20} p^6\). Simplifying exponents helps to consolidate and reduce the complexity of algebraic expressions, making them easier to handle and understand.