Problem 100 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ \frac{10 x^{3} y^{-7}}{3 x^{5} z^{2}} $$

Step-by-Step Solution

Answer

Question: Rewrite the given expression using only positive exponents: \(\frac{10 x^{3} y^{-7}}{3 x^{5} z^{2}}\) Answer: \(\frac{10z^2}{3x^2y^7}\)

Step 1: Simplify the expression

First, we need to simplify the given expression, which is: $$ \frac{10 x^{3} y^{-7}}{3 x^{5} z^{2}} $$ We can separate the components into individual fractions: $$ =\frac{10}{3} \cdot \frac{x^{3}}{x^{5}} \cdot \frac{y^{-7}}{z^{2}} $$

Step 2: Use the exponent rule for division

Now we will use the exponent rule for dividing expressions with the same base: If \(a\) is a nonzero constant and \(m\) and \(n\) are integers, then the rule is: \(a^{m} ÷ a^{n} = a^{m-n}\) Apply the rule to the \(x\) terms: $$ \frac{x^3}{x^5} = x^{3-5} = x^{-2} $$

Step 3: Eliminate the negative exponents

To eliminate the negative exponents, we will use the rule: \(a^{-m} = \frac{1}{a^{m}}\) Apply this rule to the \(x\) and \(y\) terms: $$ x^{-2} = \frac{1}{x^{2}} ,\: y^{-7} = \frac{1}{y^7} $$

Step 4: Combine the simplified terms

Now we will reassemble the expression by inserting the simplified terms into the individual fractions: $$ \frac{10}{3} \cdot \frac{1}{x^2} \cdot \frac{1}{y^7} \cdot z^2 $$

Step 5: Rewrite the expression with positive exponents

Finally, write the expression with only positive exponents by combining all the terms in a single fraction: $$ \frac{10z^2}{3x^2y^7} $$ The given expression has been rewritten using only positive exponents as: $$ \frac{10z^2}{3x^2y^7} $$

Key Concepts

Negative Exponents

Understanding negative exponents is crucial for simplifying expressions, especially when dealing with variables. A negative exponent means that the base is on the wrong side of a fraction and needs to be moved to the opposite side to become positive. For example, if you have \( a^{-m} \), this is equivalent to \( \frac{1}{a^m} \). In our exercise, \( y^{-7} \) turns into \( \frac{1}{y^7} \). This is handy when rewriting expressions to have all positive exponents, making them look neater and often simplifying calculations.
Remember that the rule applies universally:

• If the base with a negative exponent is in the numerator, move it to the denominator, and vice versa.
• This does not change the value of the expression, just its form.

Negative exponents may seem confusing at first, but with practice, they become an easy tool for expression manipulation.

Exponent Rules

Exponent rules might seem like magic at first, but they have straightforward logical foundations. The primary rules for exponents relate to their operations when adding, subtracting, multiplying, and dividing powers. When dividing two expressions with the same base, like \(\frac{x^3}{x^5}\), we utilize the division rule: \( a^{m} \div a^{n} = a^{m-n} \). Applying this rule gives us \( x^{3-5} = x^{-2} \).
Other essential rules to remember include:

• Multiplying powers with the same base increases their exponent: \(a^m \cdot a^n = a^{m+n}\).
• Raising a power to another power multiplies their exponents: \((a^m)^n = a^{m\cdot n}\).
• Any base raised to the zero power is 1: \(a^0 = 1\), as long as \(a eq 0\).

These rules are tools that allow you to simplify and manipulate expressions consistently and systematically.

Fractional Expressions

In mathematics, fractional expressions often involve variables and exponents. They can appear daunting, but understanding their structure is key. To simplify, you'll usually need to apply exponent rules and convert negative exponents into positive ones, which often changes the location of a variable in the fraction from numerator to denominator or vice versa.
For example, in the original expression \(\frac{10 x^{3} y^{-7}}{3 x^{5} z^{2}} \), each part needs to be broken down:

• Start by separating the constants from the variables: here it's \(\frac{10}{3}\).
• Apply the division rule to terms with the same base. As shown above, \( \frac{x^3}{x^5} = x^{-2} \).
• Change negative exponents into positive by flipping them across the fraction line.

Eventually, fractional expressions become more manageable, leading to a final expression like \(\frac{10z^2}{3x^2y^7} \), which uses only positive exponents and is cleaner and easier to understand. Simplifying these takes practice but is empowering once mastered.