Problem 26 - Elementary Algebra

Question

Find each value. Assume the base is not zero. $$ \frac{14 x^{r} y^{p} z^{q}}{2 x^{r} y^{h} z^{5}} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: $\frac{14 x^{r} y^{p} z^{q}}{2 x^{r} y^{h} z^{5}}$ Answer: $7 y^{p - h} z^{q - 5}$

Step 1: Divide the coefficients

First, we divide the coefficients 14 and 2 in the given expression: $$ \frac{14}{2} = 7 $$

Step 2: Subtract the exponents of x

Now, we need to subtract the exponent in the denominator (r) from the exponent in the numerator (r) for x: $$ x^{r} \div x^{r} = x^{r - r} $$ Since the exponents are the same, based on the rule of exponents, their subtraction is equal to 0: $$ x^{r - r} = x^{0} $$ Anything raised to the power of 0 is equal to 1, so: $$ x^{0} = 1 $$

Step 3: Subtract the exponents of y

Now, we will subtract the exponent in the denominator (h) from the exponent in the numerator (p) for y: $$ y^{p} \div y^{h} = y^{p - h} $$

Step 4: Subtract the exponents of z

Finally, we will subtract the exponent in the denominator (5) from the exponent in the numerator (q) for z: $$ z^{q} \div z^{5} = z^{q - 5} $$

Step 5: Combine the results

We will now combine the results obtained in Steps 1-4 to form the simplified fraction: $$ \frac{14 x^{r} y^{p} z^{q}}{2 x^{r} y^{h} z^{5}} = 7 \cdot x^{0} \cdot y^{p - h} \cdot z^{q - 5} = 7 \cdot 1 \cdot y^{p - h} \cdot z^{q - 5} $$ So the final simplified form is: $$ 7 y^{p - h} z^{q - 5} $$

Key Concepts

Exponents

Exponents are a key part of algebra and are used to express repeated multiplication of a number by itself. When simplifying algebraic expressions, understanding how to manipulate exponents is crucial.
In the process of simplification, one of the main rules used is that when you divide like bases, you subtract their exponents. This stems from the property of exponents which states that \( a^m \div a^n = a^{m-n} \).
Here's an easier breakdown to understand the rule:

• If you have two exponents with the same base, you subtract the bottom exponent from the top exponent when dividing.
• This subtraction allows us to simplify the expression to a simpler form.
• For example, \( x^r \div x^r = x^{r-r} \), which simplifies to \( x^0 \).
• An important fact to remember is that any number, except zero, raised to the power of zero is 1, hence \( x^0 = 1 \).

Understanding these rules helps us efficiently simplify expressions involving exponents.

Fraction Simplification

Simplifying fractions is a fundamental skill in solving algebraic expressions. In algebra, fractions often include variables raised to exponents, just like in our original exercise.
To simplify, you need to break the fraction down into more manageable pieces. Let's look at the step by step:

• First, simplify the coefficients (numerical parts without variables). For example, \( \frac{14}{2} = 7 \).
• Next, apply the rule of exponents for each variable term. Subtract the exponent in the denominator from the exponent in the numerator for each base because we are dividing.
• Finally, rewrite the fraction as per the simplified terms, which brings all these simplified parts together into a single expression.

Remember, simplifying fractions not only makes them easier to work with but also reveals important insights about the relationships within the equation.

Mathematical Operations

Mathematical operations are the processes that we use to perform the majority of algebraic calculations. In the exercise, we encounter operations like division of numbers and the subtraction of exponents, which are key to simplifying expressions.
Here's how these operations are applied in algebraic simplification:

• Division: When faced with a fraction, our first task is typically to divide the coefficients, simplifying them to their lowest terms.
• Subtraction of Exponents: After that, we look at the exponents on the variables. When dividing two terms with the same base, we subtract the exponent of the denominator from that of the numerator.
• Multiplication: Lastly, remember that if you end up with terms at the top of a fraction after simplification, you multiply these simplified terms together.

With these operations, expressions become less complicated and more straightforward to work with. Understanding each operation's role will enhance your algebra skills and offer insight during problem-solving processes.