Problem 72 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(\left(\frac{2}{b^{5}}\right)^{-2}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{b^{10}}{4}\).

Step 1: Apply the Negative Exponent Rule

The expression given is \( \left( \frac{2}{b^{5}} \right)^{-2} \). According to the negative exponent rule, \( a^{-n} = \frac{1}{a^n} \). Let's rewrite the expression to eliminate the negative exponent:\[\left( \frac{2}{b^{5}} \right)^{-2} = \frac{1}{\left( \frac{2}{b^{5}} \right)^2}\]

Step 2: Simplify the Expression with Positive Exponent

Now, simplify \(\left( \frac{2}{b^{5}} \right)^2\). Raising a fraction to a power means raising both the numerator and the denominator to that power:\[\left( \frac{2}{b^{5}} \right)^2 = \frac{2^2}{(b^{5})^2} = \frac{4}{b^{10}}\]

Step 3: Substitute Back into Expression

Now substitute \(\frac{4}{b^{10}}\) back into the expression from Step 1:\[\frac{1}{\frac{4}{b^{10}}} = \frac{b^{10}}{4}\]

Step 4: Present the Final Simplified Form

The expression \(\left(\frac{2}{b^{5}}\right)^{-2}\) simplifies to \(\frac{b^{10}}{4}\) with all positive exponents.

Key Concepts

Fractional Exponents

Fractional exponents are used as an alternative way to express roots, especially when dealing with powers. When you see an exponent in the form of a fraction, it indicates a relationship between powers and roots. For example, the fractional exponent \( a^{\frac{1}{n}} \) is equivalent to the \( n^{th} \) root of \( a \).
This connection helps in simplifying expressions and solving equations involving roots more systematically.

• The numerator of the fraction indicates the power.
• The denominator of the fraction signifies the root.

Fractional exponents simplify the process of calculation as they allow combining powers and roots, which is especially useful in advanced mathematics, though the underlying concept is similar to taking a square root or cube root.

Simplification

Simplification involves rewriting an expression in its simplest or most efficient form. Simplifying mathematical expressions is crucial for making calculations easier and more understandable.
In simplifying, you change the expression so that it is easier to handle while still representing the same value or statement.

• Combine like terms.
• Use operations to reduce complexity.
• Apply exponent rules correctly to reorder and reduce exponents.

In the original exercise, simplification involved managing exponents and fractions to reduce the expression to its simplest terms. This required eliminating negative exponents and rearranging components to achieve \( \frac{b^{10}}{4} \), a fraction with no negative elements.

Exponent Rules

Exponent rules help us manipulate mathematical expressions involving powers more easily. Each rule serves a different purpose and helps in different situations.
Some of the key exponent rules are:

• Product of Powers Rule: \( a^m \cdot a^n = a^{m+n} \)
• Quotient of Powers Rule: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a Power Rule: \( (a^m)^n = a^{m \cdot n} \)
• Power of a Product Rule: \( (ab)^n = a^n \cdot b^n \)
• Negative Exponent Rule: \( a^{-n} = \frac{1}{a^n} \)

Understanding and using these rules is crucial for solving expressions like \( \left( \frac{2}{b^{5}} \right)^{-2} \). The major focus was on the Negative Exponent Rule, which dictated the conversion of negative to positive exponents, making arithmetic simpler and more straightforward.