Problem 67 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ 7 a^{-3} b^{-9} \cdot 5 a^{6} b c^{-2} c^{4} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression and rewrite it with positive exponents: $(7a^{-3}b^{-9})(5a^{6}bc^{-2}c^{4})$ Answer: $\frac{35a^3c^2}{b^8}$

Step 1: Multiply the coefficients

Start by multiplying the coefficients, 7 and 5: $$\begin{aligned} (7a^{-3}b^{-9})(5a^{6}bc^{-2}c^{4}) &= (7 \times 5)(a^{-3}b^{-9})(a^{6}bc^{-2}c^{4}) \end{aligned}$$

Step 2: Combine terms with the same base

Now combine the terms with the same base by adding their exponents: $$\begin{aligned} (7 \times 5)(a^{-3}b^{-9})(a^{6}bc^{-2}c^{4}) &= 35 \cdot a^{-3+6} \cdot b^{-9+1} \cdot c^{-2+4} \end{aligned}$$

Step 3: Simplify the expression

Simplify the expression by calculating the new exponents: $$\begin{aligned} 35 \cdot a^{-3+6} \cdot b^{-9+1} \cdot c^{-2+4} &= 35 \cdot a^{3} \cdot b^{-8} \cdot c^{2} \end{aligned}$$

Step 4: Rewrite terms with negative exponents

Rewrite the term with a negative exponent using the exponent rule $$a^{-n} = \frac{1}{a^n}$$: $$\begin{aligned} 35 \cdot a^{3} \cdot b^{-8} \cdot c^{2} &= \frac{35a^3c^2}{b^8} \end{aligned}$$ Therefore, the simplified expression using only positive exponents is: $$\frac{35a^3c^2}{b^8}$$.

Key Concepts

Understanding Positive Exponents

When you see an exponent, it's like a shorthand for repeated multiplication. A positive exponent tells us how many times to multiply the base number by itself. For instance, when you have an expression like \( x^3 \), it means \( x \) multiplied by itself three times: \( x \cdot x \cdot x \).

In solving problems like the one from our exercise, using positive exponents makes expressions more intuitive and manageable because they reflect multiplication directly. On the other hand, a negative exponent such as \( a^{-3} \) implies division, as it's equivalent to \( \frac{1}{a^3} \). The concept can be visualized by 'flipping' the base with a negative exponent to the bottom of a fraction, converting it to a positive exponent.

Exponent Rules Simplified

To simplify expressions with diverse exponents, there are certain rules we must follow. Two fundamental rules are:

• The Product Rule: When multiplying with the same base, add the exponents. In mathematical form, \( a^m \times a^n = a^{m+n} \).


• The Quotient Rule: When dividing with the same base, subtract the exponents. In symbols, \( a^m \text{÷} a^n = a^{m-n} \).

Applying these rules helps in breaking down complex algebraic expressions into simpler formats. For instance, in our exercise, when we combined \( a^{-3} \) and \( a^6 \), we added the exponents to get \( a^3 \). Similarly, for bases \( b \) and \( c \), we applied the Product Rule. It is also vital to remember that any number or variable to the power of 0 is 1, simplifying expressions further if such a case arises.

Navigating Algebraic Expressions

Algebraic expressions are like puzzles involving numbers and letters where we apply known operations to solve or simplify them. They can contain variables (like \( a \), \( b \), and \( c \)), coefficients (numerical factors like 7 and 5), and exponents (like \( -3 \), \( 6 \), and \( -9 \)).

The key to solving these puzzles is to understand and apply the rules of algebra systematically. For example, to simplify an expression, we can combine like terms (terms with the same variable to the same power) and use operations appropriately. In our exercise, we multiplied coefficients together, combined like terms by adding their exponents, and finally rewrote negative exponents to positive ones to achieve a simplified expression with only positive exponents.

It's important to approach the simplification methodically to avoid errors. By breaking down the algebraic puzzle piece by piece, we can solve even the most daunting expressions with confidence.