Problem 17 - College Algebra

Question

For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents. $$ \left(12^{3} \cdot 12\right)^{10} $$

Step-by-Step Solution

Answer

The expression simplifies to \(12^{40}\).

Step 1: Apply the Product of Powers Property

The expression inside the parentheses is \(12^3 \cdot 12\). According to the Product of Powers Property, when multiplying expressions with the same base, add their exponents: \[12^3 \cdot 12^1 = 12^{3+1} = 12^4\]

Step 2: Apply the Power of a Power Property

Now that we have simplified inside the parentheses to \(12^4\), we use the Power of a Power Property. According to this property, when raising a power to another power, multiply the exponents: \[(12^4)^{10} = 12^{4 \times 10} = 12^{40}\]

Step 3: Final Expression

By applying the properties of exponents, we have rewritten the expression with a single base. The expression \((12^3 \cdot 12)^{10}\) is simplified to \[12^{40}\]

Key Concepts

Product of Powers Property

When working with exponents, understanding the product of powers property is essential. This property is useful when multiplying expressions that have the same base. Rather than multiplying the bases themselves, you simply add their exponents.
For example: If you have two expressions like \(a^m \cdot a^n\), the product of powers property allows you to combine these into a single expression: \(a^{m+n}\). This rule makes your calculations more manageable and easier to solve.

• **Example:** \(12^3 \cdot 12^1 = 12^{3+1} = 12^4\)

This illustrates how the property works by adding the exponents (3 + 1) to simplify the expression into one with a single base.

Power of a Power Property

The power of a power property helps us to simplify expressions where an exponent is being raised to another exponent. Rather than dealing with each power individually, you multiply the exponents together.
For the expression \((a^m)^n\), the property states you can multiply the exponents: \(a^{m \times n}\). This property is essential when dealing with complex powers and ensures a streamlined approach to simplifying such expressions.

• **Example:** For \((12^4)^{10}\), using the power of a power property, you multiply the exponents: \(12^{4 \times 10} = 12^{40}\)

This step simplifies the overall expression significantly, turning what could have been a complex calculation into a simple one.

Positive Exponents

In the world of mathematics, it's often important to express final answers with positive exponents. Positive exponents tell us how many times a number should be multiplied by itself.
Having a positive exponent is clearer and simpler to manipulate in further calculations. It provides a standard form that's commonly understood and preferred.

• **Why Positive Exponents Matter:** They are straightforward and eliminate confusion regarding the direction of multiplication or division.

Using positive exponents not only aligns with conventions but also helps you avoid potential errors in calculations when solving more advanced algebraic problems.