Problem 14 - Algebra 1

Question

Use the quotient of powers property to simplify the expression. $$ \left(\frac{2}{b}\right)^{4} $$

Step-by-Step Solution

Answer

The simplified form of the expression \( \left(\frac{2}{b}\right)^{4} \) is \( 16 / b^{4} \) or alternatively \( 16b^{-4} \).

Step 1: Apply the Quotient of Powers Property

We can simplify \( \left(\frac{2}{b}\right)^{4} \) by applying the quotient of powers property, which states \( \left(\frac{a}{b}\right)^{n} = a^{n} / b^{n} \). Here, \( a = 2 \), \( b = b \) (and \( b≠0 \)) and \( n = 4 \). So, \( \left(\frac{2}{b}\right)^{4} = 2^{4} / b^{4} \).

Step 2: Simplify the Expression

Calculate \( 2^{4} \) to simplify the numerator. So, \( 2^{4} = 16 \), hence the expression simplifies to \( 16 / b^{4} \) or alternatively \( 16b^{-4} \).

Key Concepts

Simplifying Expressions

Simplifying mathematical expressions is crucial to understanding and solving problems efficiently.
When dealing with expressions like \( \left(\frac{2}{b}\right)^{4} \), simplification helps to evaluate such expressions. It involves primarily reducing an expression to its most basic form while keeping it mathematically equivalent to the original.

Take the expression given: simplifying \( \left(\frac{2}{b}\right)^{4} \) involves applying exponent rules to express the exponential term in its simplest form. Once we apply the quotient of powers rule as seen in our solution, the expression transforms to a form that's easier to comprehend and utilize, resulting in \( 16b^{-4} \) or \( 16 / b^{4} \). This final expression, devoid of the exponent covering both the numerator and denominator, now appears less intimidating and more straightforward for further calculations or applications.

Exponent Rules

Understanding exponent rules, including the quotient of powers property, is vital when working with expressions that involve exponents.

In the exercise, we apply one such rule, which articulates that the quotient of two powers with the same base can be simplified by subtracting the exponents: \( \left(\frac{a}{b}\right)^{n} = a^{n} / b^{n} \).

Other Exponent Rules Include:

• Product of Powers: \( a^{m} \cdot a^{n} = a^{m+n} \).
• Power of a Power: \( (a^{m})^{n} = a^{m \cdot n} \).
• Zero Exponent: \( a^{0} = 1 \) for any non-zero \( a \).
• Negative Exponent: \( a^{-n} = 1/a^{n} \) for any non-zero \( a \) and \( n \).

By familiarizing yourselves with these rules, you become equipped to streamline the process of simplification and easily solve seemingly complex expressions.

Negative Exponents

Dealing with negative exponents can initially seem daunting, but they follow a consistent and logical rule.
If an expression contains a negative exponent, it indicates the reciprocal of the base raised to the absolute of the exponent. In other words, \( a^{-n} = 1/a^{n} \).

Applying this to our exercise where we simplify \( 16 / b^{4} \) to \( 16b^{-4} \), understanding the negative exponent tells us that \( b^{-4} \) represents taking the reciprocal of \( b \) four times, which is equivalent to dividing by \( b \) four times. This knowledge not only aids in simplification but also helps in understanding the relationships and operations that govern algebraic structures involving powers and exponents.