Problem 17 - Elementary Algebra

Question

When simplifying the terms for the following problems, write each so that only positive exponents appear. $$ \frac{\left(3^{-6}\right)\left(3^{2}\right)\left(3^{-10}\right)}{\left(3^{-5}\right)\left(3^{-9}\right)} $$

Step-by-Step Solution

Answer

Question: Simplify the following expression and rewrite it with only positive exponents: $$ \frac{(3^{-6})(3^2)(3^{-10})}{(3^{-5})(3^{-9})} $$ Answer: 1

Step 1: Use exponent laws to simplify nominator and denominator separately

We'll begin by simplifying the terms in the numerator (top) and the denominator (bottom) separately, using the properties of exponents. For the numerator, we have: $$(3^{-6})(3^2)(3^{-10}) = 3^{-6+2+(-10)} = 3^{-14}$$ For the denominator, it simplifies to: $$(3^{-5})(3^{-9}) = 3^{-5+(-9)} = 3^{-14}$$ So, the expression becomes: $$\frac{3^{-14}}{3^{-14}}$$

Step 2: Simplify the fraction using the properties of exponents

Since we have the same exponent in both the numerator and the denominator, the fraction simplifies to: $$\frac{3^{-14}}{3^{-14}} = 1$$ So, the final simplified expression is 1, with no exponents involved.

Key Concepts

Exponents

Exponents are a fundamental part of algebra, simplifying computations by representing repeated multiplication. When you see an expression like \(3^4\), it means the number 3 is multiplied by itself four times: \(3 \times 3 \times 3 \times 3\). An understanding of exponent laws helps greatly in simplifying expressions and solving equations. Here are some key rules:

• Product of Powers Rule: If you multiply like bases, you add their exponents. For example, \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers Rule: If you divide like bases, you subtract the exponents. This is seen in \(a^m / a^n = a^{m-n}\).
• Power of a Power Rule: If you raise a power to another power, you multiply the exponents: \((a^m)^n = a^{m \times n}\).
• Negative Exponent Rule: A negative exponent means division by that number of base multiplications. So, \(a^{-n} = 1/a^n\).

Recognizing and applying these rules makes it easier to work with exponents, whether simplifying expressions or solving larger algebraic problems.

Simplifying Expressions

Simplifying expressions aims to make them more straightforward without changing their value. This usually involves using algebraic rules like distributive property and exponent laws. In our exercise, to simplify it, we applied the exponent rules specifically to combine exponents both in multiplication and division of powers.
When faced with a complex expression:

• Identify like terms: Are there any variables or constants that can be combined?
• Apply the exponent laws: As shown in the step-by-step solution, combine the exponents of terms with the same base.

In the given problem, simplifying the numerator and the denominator separately allowed us to combine exponents easily. This led to expressing them as \(3^{-14}\) for both, which made further simplification much easier.
Don't forget to check for common bases since it gives you a straightforward path to reduce expression size and make calculations easier.

Fraction Simplification

Simplifying fractions is key in algebra to achieve the most manageable form of a problem. The process often involves reducing the fraction to its lowest terms. In the context of algebraic expressions with exponents, the goal is to cancel out elements and leave the simplest form possible.
Here's how you can tackle fraction simplification:

• Like Terms Above and Below: If the numerator and denominator have terms with the same base and exponent, you can cancel them, as seen in \(\frac{3^{-14}}{3^{-14}} = 1\).
• Reduction to Simplest Form: If the numerator and denominator can be divided by a common term, you should perform this division.

Always remember that once the fraction is equal in both parts, as in our solution, it simplifies to 1. No matter the complexity, simplifying fractions reduces complicated problems to simpler expressions that are much easier to handle.