Problem 16 - Algebra for College Students

Question

Simplify each numerical expression. \(3^{-4} \cdot 3^{6}\)

Step-by-Step Solution

Answer

The simplified expression is 9.

Step 1: Understanding the Base and Exponent

This expression, \(3^{-4} \cdot 3^{6}\), involves two powers with the same base, \(3\). To simplify, we use the laws of exponents, particularly the law of multiplying powers with the same base: \(a^m \cdot a^n = a^{m+n}\).

Step 2: Applying the Laws of Exponents

Using the exponent multiplication law, combine the exponents: \(3^{-4} \cdot 3^{6} = 3^{-4+6}\). This simplifies the expression to combine the exponents.

Step 3: Simplifying the Exponent

Calculate the sum of the exponents: \(-4 + 6 = 2\). Thus, the expression simplifies to \(3^2\).

Step 4: Compute the Final Value

Now that we have simplified the expression to \(3^2\), calculate the result by evaluating \(3^2 = 3 \times 3 = 9\).

Key Concepts

Multiplying Powers

In mathematics, understanding the concept of multiplying powers is crucial when dealing with exponential expressions. When multiplying powers with the same base, you can significantly simplify the expression by applying the laws of exponents. The essential rule to remember is:

• For any base, let's call it \(a\), and exponents \(m\) and \(n\), the expression \(a^m \cdot a^n\) equals \(a^{m+n}\).

This rule simplifies the otherwise complex multiplication of powers into an easily manageable operation. For example, in the exercise \(3^{-4} \cdot 3^{6}\), both terms share the base \(3\).
By applying this rule, you combine the exponents: \(-4 + 6\). Therefore, using multiplication of powers, we can quite smoothly rewrite the expression as \(3^2\), which is straightforward to evaluate.

Negative Exponents

Negative exponents can often appear daunting, but they're quite manageable once you understand their meaning. A negative exponent indicates the reciprocal of the base raised to the opposite positive power.

• For example, \(a^{-n}\) is equivalent to \(\frac{1}{a^n}\).

This doesn't mean negative numbers, but rather another way to express division involving powers.
In the example of \(3^{-4}\), it translates to \(\frac{1}{3^4}\). This provides a compact way to express fraction operations. However, when multiplying or simplifying expressions like \(3^{-4} \cdot 3^{6}\), the negative exponent gets combined with the positive via the multiplication rule, revealing the result without needing conversion into a fraction. This step - changing \(3^{-4}\) into a positive exponent \(3^2\) after combining with \(3^6\) - is crucial for simplicity and clarity.

Simplifying Expressions

Simplifying expressions involves reducing them into their simplest form while maintaining equality. This process not only makes solutions manageable but also enhances understanding of the material. When dealing with powers, simplification relies heavily on the laws of exponents.

• Start by identifying like bases and then applying the relevant exponent rules.
• In the expression \(3^{-4} \cdot 3^{6}\), the bases are the same, allowing simplification by adding the exponents: \(-4 + 6\).

This results in \(3^2\), an expression easy to compute.
Finally, the calculated value of \(3^2\) equals \(9\), providing a neat and concise answer to what initially appeared as a complex expression. Recognizing parts of an expression that can be combined is the key to mastering algebraic simplification efficiently.