Simplifying Expressions: No Negative Exponents!

by Alex Johnson 48 views

Have you ever encountered an algebraic expression riddled with negative exponents and felt a little lost? Don't worry; you're not alone! Dealing with exponents can seem tricky, but with a few key principles, you can simplify even the most complex-looking expressions. In this comprehensive guide, we'll break down the process of simplifying an expression like [36a−1b−64a3b4]−2\left[\frac{36 a^{-1} b^{-6}}{4 a^3 b^4}\right]^{-2} so that it contains only positive exponents. Let's dive in!

Understanding the Basics of Exponents

Before we tackle the main problem, let's refresh our understanding of what exponents represent and the rules that govern them. Exponents are a shorthand way of expressing repeated multiplication. For instance, x3x^3 means xâ‹…xâ‹…xx \cdot x \cdot x. The number being multiplied (in this case, x) is called the base, and the exponent (in this case, 3) indicates how many times the base is multiplied by itself.

Here are some essential rules of exponents that we'll use throughout this simplification process:

  1. Product of Powers: When multiplying powers with the same base, you add the exponents: xmâ‹…xn=xm+nx^m \cdot x^n = x^{m+n}.
  2. Quotient of Powers: When dividing powers with the same base, you subtract the exponents: xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}.
  3. Power of a Power: When raising a power to another power, you multiply the exponents: (xm)n=xmâ‹…n(x^m)^n = x^{m \cdot n}.
  4. Power of a Product: When raising a product to a power, you distribute the exponent to each factor: (xy)n=xnyn(xy)^n = x^n y^n.
  5. Power of a Quotient: When raising a quotient to a power, you distribute the exponent to both the numerator and the denominator: (xy)n=xnyn\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}.
  6. Negative Exponent: A negative exponent indicates a reciprocal: x−n=1xnx^{-n} = \frac{1}{x^n}.
  7. Zero Exponent: Any non-zero number raised to the power of zero is 1: x0=1x^0 = 1 (where x≠0x \ne 0).

Why are these rules important? They provide the foundation for manipulating and simplifying expressions. By mastering these rules, you can confidently navigate complex algebraic problems. We'll see these rules in action as we simplify the given expression.

Step-by-Step Simplification of [36a−1b−64a3b4]−2\left[\frac{36 a^{-1} b^{-6}}{4 a^3 b^4}\right]^{-2}

Now, let's break down the simplification of the expression [36a−1b−64a3b4]−2\left[\frac{36 a^{-1} b^{-6}}{4 a^3 b^4}\right]^{-2} step-by-step. Our goal is to eliminate negative exponents and express the final result in its simplest form. Remember, the key is to apply the rules of exponents systematically.

Step 1: Simplify the Fraction Inside the Brackets

First, we focus on simplifying the fraction within the brackets. This involves dealing with both the numerical coefficients and the variables with their respective exponents.

36a−1b−64a3b4\frac{36 a^{-1} b^{-6}}{4 a^3 b^4}

Divide the numerical coefficients: 364=9\frac{36}{4} = 9

Now, apply the Quotient of Powers rule to the variables:

  • For a: a−1a3=a−1−3=a−4\frac{a^{-1}}{a^3} = a^{-1-3} = a^{-4}
  • For b: b−6b4=b−6−4=b−10\frac{b^{-6}}{b^4} = b^{-6-4} = b^{-10}

So, the simplified fraction inside the brackets becomes:

9a−4b−109 a^{-4} b^{-10}

Step 2: Apply the Outer Exponent

Next, we raise the simplified expression to the power of -2. This means applying the Power of a Product rule and the Power of a Power rule.

(9a−4b−10)−2(9 a^{-4} b^{-10})^{-2}

Distribute the exponent -2 to each factor:

9−2(a−4)−2(b−10)−29^{-2} (a^{-4})^{-2} (b^{-10})^{-2}

Now, apply the Power of a Power rule:

  • 9−2=192=1819^{-2} = \frac{1}{9^2} = \frac{1}{81}
  • (a−4)−2=a(−4)(−2)=a8(a^{-4})^{-2} = a^{(-4)(-2)} = a^8
  • (b−10)−2=b(−10)(−2)=b20(b^{-10})^{-2} = b^{(-10)(-2)} = b^{20}

Step 3: Combine the Terms

Now, we combine the simplified terms:

181a8b20\frac{1}{81} a^8 b^{20}

This can also be written as:

a8b2081\frac{a^8 b^{20}}{81}

Final Answer

Therefore, the simplified form of [36a−1b−64a3b4]−2\left[\frac{36 a^{-1} b^{-6}}{4 a^3 b^4}\right]^{-2} without negative exponents is a8b2081\frac{a^8 b^{20}}{81}.

Key Strategies for Simplifying Exponential Expressions

To effectively simplify expressions with exponents, keep these strategies in mind:

  • Work Inside Out: Start by simplifying the innermost parts of the expression, such as fractions within brackets.
  • Apply Rules Systematically: Follow the rules of exponents methodically, one step at a time. This minimizes errors.
  • Deal with Negative Exponents Last: It's often easier to simplify other parts of the expression before dealing with negative exponents. Move terms with negative exponents in the final step.
  • Double-Check Your Work: After each step, review your work to ensure you haven't made any mistakes.

By consistently applying these strategies, you'll become more proficient at simplifying exponential expressions.

Common Mistakes to Avoid

When working with exponents, it's easy to make common mistakes. Being aware of these pitfalls can help you avoid them:

  • Incorrectly Applying the Quotient Rule: Ensure you subtract the exponents correctly when dividing powers with the same base. For example, x5x2\frac{x^5}{x^2} is x5−2=x3x^{5-2} = x^3, not x5/2x^{5/2}.
  • Forgetting to Distribute the Exponent: When raising a product or quotient to a power, remember to distribute the exponent to every factor. For instance, (2xy)3(2xy)^3 is 23x3y3=8x3y32^3 x^3 y^3 = 8x^3y^3, not 2x3y32x^3y^3.
  • Misunderstanding Negative Exponents: A negative exponent indicates a reciprocal, not a negative number. For example, x−2x^{-2} is 1x2\frac{1}{x^2}, not −x2-x^2.
  • Adding Exponents When Bases are Different: You can only add exponents when multiplying powers with the same base. For example, x2â‹…y3x^2 \cdot y^3 cannot be simplified further.

Practice Problems

To solidify your understanding, try simplifying these expressions:

  1. 15x4y−23x−1y5\frac{15x^4y^{-2}}{3x^{-1}y^5}
  2. (2a2b−3)4(2a^2b^{-3})^4
  3. [25p−3q25p2q−1]−1\left[\frac{25p^{-3}q^2}{5p^2q^{-1}}\right]^{-1}

Solutions:

  1. 5x5y−7=5x5y75x^5y^{-7} = \frac{5x^5}{y^7}
  2. 16a8b−12=16a8b1216a^8b^{-12} = \frac{16a^8}{b^{12}}
  3. [5p−5q3]−1=5−1p5q−3=p55q3\left[5p^{-5}q^3\right]^{-1} = 5^{-1}p^5q^{-3} = \frac{p^5}{5q^3}

Conclusion

Simplifying expressions with exponents might seem daunting at first, but with a solid understanding of the rules and consistent practice, you can master this skill. Remember to work systematically, apply the rules correctly, and double-check your work. By following these guidelines, you'll be able to confidently tackle any expression with exponents and express it in its simplest form with positive exponents.

For further exploration and practice, consider visiting reputable mathematics resources such as Khan Academy's Algebra I section for additional lessons and exercises on exponents and algebraic expressions. Happy simplifying!