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Chapter 6: Problem 98

In Exercises 97-100, simplify the expression. $$ \left(\frac{b^{-4}}{4 a^{-5}}\right)^{2} $$

Short Answer

Expert verified
The simplified expression is \( \frac{1}{4b^{8}a^{10}} \)

Step by step solution

01

Apply Power Rule

Apply the power rule, which states that \( (a^{m/n})^{p/q} = a^{(m*p)/(n*q)} \), to both terms in the brackets. The expression will then look like this: \[ \left(b^{-4 * 2}/4 * a^{-5 * 2}\right) \]
02

Evaluate Exponents

Evaluate the exponents. This simplifies the expression to: \[ \left(\frac{b^{-8}}{4a^{-10}}\right) \]
03

Move terms to Simplify

Recall the rule that \( a^{-n} = 1/a^{n} \). Therefore, \( b^{-8} = 1/b^{8} \) and \( a^{-10} = 1/a^{10} \). Substitute these values back into the equation: \[ \frac{1}{b^{8} * 4a^{10}} \]
04

Simplify the Expression

Finally, rearrange the equation to simplify the expression: \[ \frac{1}{4b^{8}a^{10}} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponents and Powers
Understanding exponents and powers is crucial for simplifying algebraic expressions. An exponent indicates how many times a number, known as the base, is multiplied by itself. For example, in the expression \( a^n \), \( a \) is the base, and \( n \) is the exponent. This tells us to multiply \( a \) by itself \( n \) times.
When you have an expression like \( (b^{-4})^2 \), you apply the power rule. The power rule for exponents says that you multiply the exponents together. So, \( (b^{-4})^2 = b^{-4 \times 2} = b^{-8} \).
  • Base: The number being multiplied.
  • Exponent: The number of times the base is multiplied by itself.
  • Power Rule: Multiply the exponents when raising a power to another power.
Negative Exponents
Negative exponents might seem tricky at first, but they're easy once you get the hang of it. A negative exponent tells us to take the reciprocal of the base and make the exponent positive.
For instance, \( b^{-8} \) can be rewritten as \( \frac{1}{b^8} \). This rule makes it possible to move terms with negative exponents from the numerator to the denominator and vice versa, simplifying the expression in the process.
It's essential to remember:
  • Reciprocal: It flips the base from the numerator to the denominator or vice versa.
  • Negative Exponent Rule: \( a^{-n} = \frac{1}{a^n} \).
Applying these rules is crucial for simplifying expressions effectively.
Algebraic Fractions
Algebraic fractions look daunting when combined with exponents, but they're manageable by following basic fraction rules. An algebraic fraction is simply a fraction with variables in the numerator, denominator, or both.
To simplify algebraic fractions, follow these steps:
  • Evaluate Exponents First: Simplify any exponents using the power and negative exponent rules.
  • Simplify the Fraction: Combine like terms and simplify any coefficients, as you would with regular fractions.
A fraction like \( \frac{b^{-8}}{4a^{-10}} \) requires us to apply the negative exponent rule first, rewriting it as \( \frac{1}{4b^8a^{10}} \).
Simplifying algebraic fractions involves:
  • Rewriting expressions with positive exponents.
  • Combining like terms to simplify the fraction.
Understanding these steps is key to mastering algebraic fractions.

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