Problem 96 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ (5 \cdot 2)^{5} $$

Step-by-Step Solution

Answer

The simplified expression is '100,000'.

Step 1: Identify and Apply the Power of a Product Rule

In the expression \((5 \cdot 2)^{5}\), the power of a product rule applies. This rule states that \((ab)^n = a^n \cdot b^n\). Thus, the expression simplifies into \(5^5 \cdot 2^5\).

Step 2: Calculate Each Exponent Individually

Next is to calculate the results of \(5^5\) and \(2^5\), respectively. These will be '3125' and '32'.

Step 3: Multiply the Results

Lastly, multiply '3125' and '32' to get the final simplified value, which is '100,000'.

Key Concepts

Power of a Product

When dealing with the expression \((5 \cdot 2)^5\), the Power of a Product rule is helpful. This mathematical rule lets us distribute the power to each factor inside the parentheses. Simply put, it states that \((ab)^n = a^n \cdot b^n\). This means we should apply the exponent to both '5' and '2' separately.

The beauty of this rule is that it simplifies calculations. Instead of multiplying the factors and then raising to the power, the rule allows for individual exponentiation of each factor first, which often makes mental math and computation easier.

• Apply the exponent to each factor: \((5 \cdot 2)^5\) becomes \(5^5 \cdot 2^5\).
• Remember this rule whenever you see a product inside parentheses with an exponent outside.

This ensures that you stay organized and prevent mistakes by misapplying the exponent.

Simplifying Expressions

Simplifying expressions is a fundamental skill in mathematics. It involves reducing an expression to its most basic form without changing its value. For \((5 \cdot 2)^5\), simplification involves using rules such as the Power of a Product to make the expression easier to work with.

First, identify if the expression can be altered using mathematical rules like distributing an exponent or combining like terms.

• Use the Power of a Product: Distribute the exponent to each term within the parenthesis.
• Break expressions into simpler parts: \(5^5\) and \(2^5\) are simpler and easier to calculate individually.
• Re-check your work: Each step should maintain the equivalence of the original expression.

Ultimately, simplifying expressions provides clarity and makes complex calculations more manageable.

Multiplication of Exponents

Once you have simplified an expression by applying relevant rules, you'll often need to perform operations involving exponents. In \(5^5 \cdot 2^5\), calculate each exponent individually.

The steps are straightforward but crucial:

• Calculate \(5^5\): Multiply '5' by itself five times, resulting in '3125'.
• Calculate \(2^5\): Do the same with '2', leading to '32'.
• Multiply the results: Finally, perform the multiplication, '3125 \cdot 32', which equals '100,000'.

By breaking the problem into smaller parts and handling each exponent separately, you manage the complexity and succeed in finding the result efficiently. These strategies ensure that you maintain accuracy while performing calculations.