Problem 46 - Elementary Algebra

Question

Write \(\left(\frac{15 x^{-3} y^{4}}{5 x^{2} y^{-7}}\right)^{-2}\) so that only positive exponents appear.

Step-by-Step Solution

Answer

Question: Simplify the expression \(\left(\frac{15 x^{-3} y^{4}}{5 x^{2} y^{-7}}\right)^{-2}\) so that only positive exponents appear. Answer: \(\frac{1}{9}x^2y^6\)

Step 1: Apply the power rule to the whole expression

Raise the entire expression to the power of -2, using the power of a quotient rule, which states that \({(a/b)^n = a^n/b^n}\): \(\left(\frac{15 x^{-3} y^{4}}{5 x^{2} y^{-7}}\right)^{-2} = \frac{(15 x^{-3} y^{4})^{-2}}{(5 x^{2} y^{-7})^{-2}}\)

Step 2: Apply the power rule to the numerator and denominator

Now, apply the power of a product rule, which states that \({(ab)^n = a^n b^n}\), to both the numerator and the denominator: \(\frac{(15 x^{-3} y^{4})^{-2}}{(5 x^{2} y^{-7})^{-2}} = \frac{15^{-2} (x^{-3})^{-2} (y^{4})^{-2}}{5^{-2} (x^{2})^{-2} (y^{-7})^{-2}}\)

Step 3: Apply the negative exponent rule

For each term, apply the negative exponent rule, which states that \(a^{-n} = \frac{1}{a^n}\): \(\frac{15^{-2} (x^{-3})^{-2} (y^{4})^{-2}}{5^{-2} (x^{2})^{-2} (y^{-7})^{-2}} = \frac{\frac{1}{15^2}\frac{1}{(x^{-3})^2}\frac{1}{(y^{4})^2}}{\frac{1}{5^2}\frac{1}{(x^{2})^2}\frac{1}{(y^{-7})^2}}\)

Step 4: Simplify terms

Simplify the terms of the expression: \(\frac{\frac{1}{15^2}\frac{1}{(x^{-3})^2}\frac{1}{(y^{4})^2}}{\frac{1}{5^2}\frac{1}{(x^{2})^2}\frac{1}{(y^{-7})^2}} = \frac{1}{225}\cdot\frac{x^6}{1}\cdot\frac{1}{y^8}\cdot\frac{25}{1}\cdot\frac{1}{x^4}\cdot\frac{y^{14}}{1}\)

Step 5: Combine the terms

Combine the terms to obtain the simplified expression with only positive exponents: \(\frac{1}{225}\cdot\frac{x^6}{1}\cdot\frac{1}{y^8}\cdot\frac{25}{1}\cdot\frac{1}{x^4}\cdot\frac{y^{14}}{1} = \frac{25x^6y^{14}}{225x^4y^8}\)

Step 6: Simplify the coefficients and exponents

Simplify the coefficients and exponent values by dividing the coefficients of the numerator by the coefficients of the denominator and subtract the exponent in the denominator from the exponent in the numerator for each variable: \(\frac{25x^6y^{14}}{225x^4y^8} = \frac{25}{225}x^{6-4}y^{14-8} = \frac{1}{9}x^2y^6\) The given expression is now simplified with only positive exponents and its simplified form is \(\frac{1}{9}x^2y^6\).

Key Concepts

Power Rule

The power rule is a foundational concept in algebra and exponentiation, vital for simplifying expressions with exponents. It essentially provides a guideline for managing exponents when a power is raised to another power. The rule states:

• For a single term: \((a^m)^n = a^{m imes n}\).
• For a quotient (fraction):\((\frac{a}{b})^n = \frac{a^n}{b^n}\).

In practice, when you see an expression that looks like a fraction raised to an exponent, apply the power to both the numerator and the denominator. For example, in the given exercise, the entire fraction is raised to the power of \(-2\). Therefore, by applying the power rule, we have:\[\left(\frac{15 x^{-3} y^{4}}{5 x^{2} y^{-7}}\right)^{-2} = \frac{(15 x^{-3} y^{4})^{-2}}{(5 x^{2} y^{-7})^{-2}}\].Each part of the fraction (numerator and denominator) is raised to the power of \(-2\), simplifying the computation.

Negative Exponent Rule

Understanding negative exponents is crucial as they often appear intimidating but are quite straightforward with the right approach.The negative exponent rule states that:

• For any non-zero number, \(a^{-n} = \frac{1}{a^n}\).This essentially means we take the reciprocal of the base raised to the opposite positive power.
  • If \(x^{-3}\) is part of your expression, it transforms to \(\frac{1}{x^3}\).

So, using the negative exponent rule, the terms in the expression \(\left(\frac{15 x^{-3} y^{4}}{5 x^{2} y^{-7}}\right)^{-2}\) are changed such that all negative exponents convert into positive ones by flipping their positions:

• From the exercise: \( (x^{-3})^{-2} = x^6 \)
• and, \((y^{-7})^{-2} = y^{14}\).

By restructuring these negative powers, the expression becomes easier to simplify further.

Simplifying Expressions

Simplifying expressions is the process of transforming an equation into its simplest and most compact form.It's about making an expression easy to understand and work with by using algebraic rules efficiently.First, ensure all exponents are positive, leveraging both the power rule and negative exponent rule previously explained.
With the expression \(\frac{(15 x^{-3} y^{4})^{-2}}{(5 x^{2} y^{-7})^{-2}}\), we simplify by doing the following:

• Compute all powers as needed using the power rule.
• Convert negative exponents using \(a^{-n} = \frac{1}{a^n}\).
• Reduce or eliminate any redundancy in the coefficients and similar base variables.

For example, applying these steps leads to:\[\frac{1}{225} \cdot \frac{25}{1} \cdot \frac{x^6}{x^4} \cdot y^{14-8}\].Finally, compute the numeric part by simplifying fractions \(\frac{25}{225} = \frac{1}{9}\) and subtract exponents for like bases, resulting in just positive exponents throughout the expression:\[\frac{1}{9}x^2y^6\].This completion showcases the streamlined, positive-exponent form of the original expression.