Problem 17 - Elementary Algebra

Question

\(\frac{32 h^{9}}{8 h^{4}}\)

Step-by-Step Solution

Answer

4h^5

Step 1 - Simplify the Constant Coefficients

First, simplify the constant coefficients in the fraction. The constant terms are 32 and 8. By dividing 32 by 8, we get 4. So, the simplified coefficient is 4.

Step 2 - Simplify the Variables

Now, simplify the variables. You have \(h^9\) in the numerator and \(h^4\) in the denominator. According to the laws of exponents, \( \frac{h^9}{h^4} = h^{9-4} = h^5 \).

Step 3 - Combine the Results

Combine the simplified constant coefficient and the simplified variable. The final simplified form is \[ 4h^5 \].

Key Concepts

algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. These symbols represent quantities without fixed values, known as variables. In algebra, we often work with equations or expressions to find unknown values.

Common terms in algebra include:

• **Variables:** Symbols that represent unknown numbers, such as \( h \).
• **Constants:** Numbers that have a fixed value, like 32 or 8.
• **Expressions:** Combinations of variables and constants. For instance, \( 32h^9 \) is an expression.
• **Equations:** Statements that two expressions are equal. For example, \( 3x + 5 = 11 \).

In this exercise, we simplify an algebraic fraction. This fraction includes both a constant coefficient and variables. By understanding and applying algebraic rules, we can simplify complex expressions step by step.

laws of exponents

The laws of exponents are essential for simplifying expressions involving powers. When we deal with exponents, the following rules come in handy:

• **Product Rule:** \( a^m \times a^n = a^{m+n} \)
• **Quotient Rule:** \( \frac{a^m}{a^n} = a^{m-n} \)
• **Power of a Power Rule:** \( (a^m)^n = a^{m \times n} \)
• **Zero Exponent Rule:** \( a^0 = 1 \) for any nonzero \(a\)
• **Negative Exponent Rule:** \( a^{-n} = \frac{1}{a^n} \)

In our exercise, we use the quotient rule to simplify the variables. We started with \( \frac{h^9}{h^4} \). By applying the quotient rule, we subtract the exponents: \( 9 - 4 \), resulting in \( h^5 \).

This step illustrates how exponent rules can make complex expressions easier to manage and simplifies the algebraic fraction fundamentally.

simplifying expressions

Simplifying expressions means making a mathematical expression as simple as possible. This often involves combining like terms, reducing fractions, or applying mathematical laws (like the laws of exponents).

In our given exercise:

1. **Simplify the Constant Coefficient:** We first reduce \( \frac{32}{8} \) to 4. This reduction makes the numbers easier to work with.
2. **Simplify the Variables:** Using the laws of exponents, we simplify \( \frac{h^9}{h^4} \) to \( h^5 \). Subtracting the exponents, as shown, gives us the simplified form of the variable part.
3. **Combine Results:** The final simplified expression is \( 4h^5 \), combining both the simplified constants and variables.

Remember, the goal of simplifying is to reduce the expression to its simplest form without changing its value. Simplified expressions are easier to understand, use, and solve.