Problem 45 - Algebra A Combined Function

Question

Simplify each expression. Write each result using positive exponents only. $$ \frac{6 x^{2} y^{3} z^{0}}{-7 x^{2} y^{5} z^{5}} $$

Step-by-Step Solution

Answer

The simplified expression is \( -\frac{6}{7 y^{2} z^{5}} \).

Step 1: Simplify the Expression

First, we observe the expression \( \frac{6 x^{2} y^{3} z^{0}}{-7 x^{2} y^{5} z^{5}} \). We recognize that \( z^{0} \) is equal to 1, and thus can be removed from the numerator. We are left with the expression \( \frac{6 x^{2} y^{3}}{-7 x^{2} y^{5} z^{5}} \).

Step 2: Divide Like Terms in Numerator and Denominator

Next, we divide each matching variable in the numerator and the denominator of the fraction. For \( x^{2} \), we have \( x^{2} \) in the numerator and the denominator, so they cancel each other out. For \( y^{3} \) and \( y^{5} \), we subtract the exponent in the denominator from that in the numerator, resulting in \( y^{-2} \). For \( z^{5} \), since it's only in the denominator, we keep it as \( z^{5} \) in the denominator after simplification. This gives us the expression \( \frac{6}{-7 y^{2} z^{5}} \).

Step 3: Simplify the Coefficients

Observe the coefficients \( 6 \) and \( -7 \). They cannot be further simplified, so our final expression remains \( \frac{6}{-7 y^{2} z^{5}} \).

Step 4: Simplify the Negative Denominator

Having a negative sign in the denominator is equivalent to having it in front of the entire fraction. Thus, we rewrite the expression as \( -\frac{6}{7 y^{2} z^{5}} \). This gives us a simplified expression with only positive exponents.

Key Concepts

Simplifying Fractions

When simplifying fractions, the goal is to make the fraction as simple as possible while maintaining its value. This involves looking at both the numbers (coefficients) and the variables. To simplify:

• Check if numbers can be divided into each other.
• Cancel out common factors both in the numerator and denominator.
• Look carefully at the variables and their exponents.

For the expression \[\frac{6 x^{2} y^{3} z^{0}}{-7 x^{2} y^{5} z^{5}},\]you need to simplify it by dividing like terms and reducing any unnecessary factors, such as removing any terms with an exponent of zero, since \(z^0 = 1\).Notice that when dividing like terms, subtract the exponents of matching variables. For example, if you have \(x^2\) on both the top and bottom, they cancel out, simplifying the fraction. Make sure the final expression only includes necessary elements.

Exponent Rules

Understanding exponent rules is essential when working with algebraic expressions, especially when simplifying them. Here are the key rules:

• Product of powers: Add exponents when multiplying like bases, \(a^m \times a^n = a^{m+n}\).
• Quotient of powers: Subtract exponents when dividing like bases, \(a^m \div a^n = a^{m-n}\).
• Zero exponent: Any non-zero base raised to the power of zero is 1, \(a^0 = 1\).

Remember, when simplifying fractions, it is common to subtract the exponents of similar bases in the numerator from those in the denominator. This is the rule applied when simplifying \(y^3\) and \(y^5\) to become \(y^{-2}\). Understanding these rules helps in reducing expressions to their simplest form.

Positive Exponents

In algebra, it's often required to express results using positive exponents. Positive exponents are easy to interpret, as they represent the number of times a base is multiplied by itself. To convert negative exponents into positive ones:

• If a term with a negative exponent is in the denominator, move it to the numerator and change the exponent to positive, and vice versa.
• Negative exponent: \(a^{-n} = \frac{1}{a^n}\).

In our simplified fraction:\[y^{-2}\]and\[\frac{1}{y^2}\]represent the same value but with positive exponents. By applying these conversions, the final expression of \( -\frac{6}{7 y^{2} z^{5}} \) contains only positive exponents, making it more standard in mathematical expressions.

Variables in Algebra

Variables in algebra represent unknown or changeable values within expressions and equations. When working with variables tied to exponents, they enable the expression of powers and roots, showing relationships or simplifying expressions. The expression:\[\frac{6 x^2 y^3 z^0}{-7 x^2 y^5 z^5}\]contains various variables like \(x\), \(y\), and \(z\) with different exponents. The variable \(z\) has an exponent of 0 in the numerator, which allows it to be simplified to 1, essentially removing it. When dealing with variables:

• Identify like terms to combine or cancel them out if possible.
• Especially note variables with an exponent of 0, as they simplify straightforwardly.

Variables help form formulas and expressions and, when managed correctly, allow proper simplification and problem-solving in algebra.