Problem 46 - Algebra 1

Question

Write your answer as a power or as a product of powers. $$ \left(5 b^{2}\right)^{3}\left(\frac{1}{2} b^{3}\right)^{2} $$

Step-by-Step Solution

Answer

The simplified answer in terms of powers is \(31.25*b^{12}\)

Step 1: Applying power of a power property

Start by applying the power of a power property, which gives \((5^3) * (b^{2*3}) = 125 * b^6\). Similarly, apply this rule to the next bracket yielding \((\(\frac{1}{2}\)^2 * b^{3*2}) = \(\frac{1}{4}\) * b^6\)

Step 2: Applying power of a product property

Now use the power of a product property to multiply both results. This results in \(125*b^6 * \(\frac{1}{4}\)*b^6 = (125*\(\frac{1}{4}\)) * b^{6 + 6} = 31.25*b^{12}\)

Step 3: Final simplification

After simplifying, the expression is reduced to \(31.25*b^{12}\)

Key Concepts

Power of a Power Property

When dealing with exponents, the power of a power property is incredibly helpful for simplification. This property states that when you raise a power to another power, you multiply the exponents. For example, if you have

• \( (a^m)^n \)

this results in \( a^{m \times n} \).

In the exercise, we applied this property to both

• \( (5 b^{2})^{3} \)
• \( \left(\frac{1}{2} b^{3}\right)^{2} \)

By applying the power of a power property, we transformed

• \( b^{2} \) raised to \( 3 \) as \( b^{2 \times 3} = b^6 \)
• \( b^{3} \) raised to \( 2 \) as \( b^{3 \times 2} = b^6 \)

This results in simplified individual terms for easier calculation in subsequent steps.

Power of a Product Property

Power of a product property comes in handy when dealing with an expression that includes a product being raised to a power. This property allows you to apply the exponent to each factor within the parentheses separately. As a formula, it can be written as:

• \( (ab)^n = a^n \cdot b^n \)

In the given exercise, this property was applied first in breaking down

• \( (5b^{2})^{3} \)
• \( (\frac{1}{2}b^{3})^{2} \)

into individual components:

• \( 5^3 \cdot b^6 \)
• \( (\frac{1}{2})^2 \cdot b^6 \)

By handling each component separately, it became possible to simplify the expression further leading to accurate calculations step by step.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra allowing complex expressions to be reduced into simpler forms. In the exercise, we combined the previous steps to ultimately simplify the initial expression.

• First, by multiplying the numeric values: \( 125 \times \frac{1}{4} = 31.25 \)
• Next, by applying the properties of exponents on like terms, \( b^{6+6} = b^{12} \)

This final step showcased the utility of combining principles to reduce an expression neatly. The result in the exercise was a cleaner, more manageable form: \( 31.25 \cdot b^{12} \). Always pay attention to these rules as they convert otherwise cumbersome expressions into straightforward ones.