Problem 67 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. $$ \left(b^{2}\right)^{-4} $$

Step-by-Step Solution

Answer

\( \left(b^{2}\right)^{-4} = \frac{1}{b^8} \).

Step 1: Understanding Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the positive of that exponent. For example, \( x^{-n} = \frac{1}{x^{n}} \). Here, \( (b^2)^{-4} \) can be rewritten as \( \frac{1}{(b^2)^{4}} \).

Step 2: Applying the Power Rule

The power rule states that \( (a^m)^n = a^{m \cdot n} \). Applying this rule, we rewrite \( (b^2)^4 \) as \( b^{2 \cdot 4} = b^8 \).

Step 3: Final Expression

After applying the rules above, the expression \( \left(b^{2}\right)^{-4} \) simplifies to \( \frac{1}{b^8} \).

Key Concepts

Understanding Negative Exponents

Negative exponents can be a bit confusing at first, but once you get the hang of it, they become quite intuitive. The concept boils down to taking the reciprocal of the base. An exponent tells you how many times to multiply the base by itself. When the exponent is negative, it indicates that the base number is in the denominator as a reciprocal.
For example, if you have a base with an exponent of -1, say, \( x^{-1} \), this is equivalent to \( \frac{1}{x} \). Do you see the pattern here? So, when you encounter an expression like \( (b^2)^{-4} \), you convert it to \( \frac{1}{(b^2)^4} \). You’ve just used the negative exponent rule! This conversion is crucial for simplifying expressions in algebra.
Remember:

• Negative exponents indicate division, or putting the base in the denominator.
• Convert \( x^{-n} \) into \( \frac{1}{x^n} \).

Reciprocal Simplification

Reciprocal is a term that often pops up when dealing with negative exponents, as it represents the multiplicative inverse of a number. In simpler terms, it is what you need to multiply a number by to get 1.
For instance, the reciprocal of \( a \) is \( \frac{1}{a} \), and vice versa. Reciprocal simplification becomes practical when you're dealing with expression manipulation in algebra, especially when you need to maintain the equivalence of expressions while simplifying.

• The reciprocal of a fraction is obtained by swapping the numerator and the denominator.
• For negative exponents, change \( x^{-n} \) to its reciprocal \( \frac{1}{x^n} \).

When you rewrite \( (b^2)^{-4} \) as \( \frac{1}{(b^2)^4} \), you're employing the reciprocal concept. This simplification helps keep expressions free of negative exponents and maintains the mathematical balance.

Mastering the Power Rule

The power rule is a powerful tool in mathematics for dealing with exponents. This rule focuses on simplifying expressions where exponents themselves have exponents.
The rule is straightforward: when you have an expression of the form \( (a^m)^n \), it is equivalent to \( a^{m \cdot n} \). You multiply the exponents to simplify the expression.

• Helps in transforming complex expressions into easier forms.
• Make sure to multiply the exponents correctly to avoid errors.

Using the power rule is essential when you tackle complex exponential expressions like \( (b^2)^4 \). Here, you apply the rule to get \( b^{2 \cdot 4} = b^8 \). It significantly simplifies calculations and helps maintain accuracy. Always remember to express the final result without negative exponents, ensuring clarity and correctness.