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Magazine Chapter 11: Problem 27
Evaluate the given expressions. $$\frac{16^{-1.25}}{5}+\frac{2^{-0.6}}{2^{1.4}}$$
Short Answer
Expert verified
The result is \(\frac{41}{160}\).
Step by step solution
01
Simplify the Exponential Expressions
The expression involves powers of 16 and 2. First, rewrite the expressions using exponential rules. Starting with \(16^{-1.25}\), note that 16 can be written as \(2^4\), so:\[16^{-1.25} = (2^4)^{-1.25} = 2^{-5}\]For the second part, \(\frac{2^{-0.6}}{2^{1.4}}\), we use the property that \(\frac{a^m}{a^n} = a^{m-n}\):\[2^{-0.6 - 1.4} = 2^{-2}\]
02
Plug in Simplified Expressions
Substitute the simplified expressions back into the overall equation:\[\frac{16^{-1.25}}{5} + \frac{2^{-0.6}}{2^{1.4}} = \frac{2^{-5}}{5} + 2^{-2}\]
03
Calculate Each Term
Compute \(2^{-5}\) and \(2^{-2}\):\[2^{-5} = \frac{1}{2^5} = \frac{1}{32}\]\[2^{-2} = \frac{1}{2^2} = \frac{1}{4}\]Insert these values back into the expression:\[\frac{1/32}{5} + \frac{1}{4}\]
04
Evaluate the First Term
Calculate \(\frac{1/32}{5}\):\[\frac{1}{32} \div 5 = \frac{1}{32 \times 5} = \frac{1}{160}\]
05
Add the Two Terms Together
Finally, add \(\frac{1}{160}\) and \(\frac{1}{4}\):Find a common denominator, which is 160:\[\frac{1}{4} = \frac{40}{160}\]So,\[\frac{1}{160} + \frac{40}{160} = \frac{41}{160}\]
06
Final Step: Conclusion
The final simplified result of the expression is \(\frac{41}{160}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponentiation Rules
Exponentiation is a mathematical operation that involves raising a number, called the base, to a power, known as the exponent. Understanding the rules of exponentiation is key in simplifying complex expressions. For basic rules:
- When multiplying same bases, you add the exponents: \( a^m \times a^n = a^{m+n} \).
- When dividing same bases, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
- A power of a power requires multiplying the exponents: \( (a^m)^n = a^{m\times n} \).
- Any number raised to the zero power is one: \( a^0 = 1 \), provided \( a eq 0 \).
Fraction Simplification
Simplifying fractions is an essential part of solving algebraic expressions. To simplify a fraction means to make it as simple as possible. The process often involves reducing the numerator and denominator by their greatest common divisor. Consider:
- A basic way to simplify is dividing both the numerator and the denominator by their \textbf{greatest common factor (GCF)}.
- In more complex cases, such as those encountered in algebra, fraction simplification might also involve applying exponents and understanding their relationship in a fraction, as in \( \frac{2^{-0.6}}{2^{1.4}} = 2^{-2} \).
Common Denominators
When adding or subtracting fractions, a common denominator is necessary. This means that the denominators of the fractions must be made equal so that the numerators can be added or subtracted straightforwardly.
- The easiest way to find a common denominator is to use the \textbf{least common multiple (LCM)} of the denominators.
- Once you have a common denominator, convert each fraction: for example, \( \frac{1}{4} \) becomes \( \frac{40}{160} \) to match the common denominator of 160.
- After conversion, add or subtract the numerators while keeping the denominator constant.
