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Magazine Chapter 1: Problem 5
Express the number in the form a/b, where \(a\) and \(b\) are integers. $$-2^{4}+3^{-1}$$
Short Answer
Expert verified
The expression is equivalent to \(-\frac{47}{3}\).
Step by step solution
01
Calculate Exponentiation
First, evaluate the expression involving the exponent. Calculate \(-2^4\). This implies we should compute \(-1\times 2^4\), as the negative sign applies after the exponentiation, resulting in \(-16\).
02
Evaluate the Reciprocal
Next, evaluate the expression \(3^{-1}\). This means taking the reciprocal of 3, which gives \(\frac{1}{3}\).
03
Combine Terms
Now combine the terms \(-16\) and \(\frac{1}{3}\) by converting \(-16\) into a fraction. This results in \(-\frac{16}{1}+\frac{1}{3}\).
04
Find Common Denominator
To add the fractions, find a common denominator. The least common denominator for 1 and 3 is 3. Rewrite \(-\frac{16}{1}\) as \(-\frac{48}{3}\).
05
Add Fractions
Add the fractions \(-\frac{48}{3}+\frac{1}{3}\). Adding these gives \(-\frac{48}{3}+\frac{1}{3}= -\frac{47}{3}\).
06
Final Answer
The expression \(-2^4+3^{-1}\) can be expressed as the fraction \(-\frac{47}{3}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponentiation
In arithmetic, exponentiation is a fundamental operation that involves raising a number to the power of an exponent. This means repeatedly multiplying the base by itself. For instance, in the expression \(-2^4\), the base is \(2\) and the exponent is \(4\). Here's how it works step by step:
First, calculate \(2^4\), which means multiplying \(2\) four times: \(2 \times 2 \times 2 \times 2 = 16\).
Next, apply the negative sign in front of the base after evaluating the exponent. Therefore, \(-2^4\) is computed as \(-1 \times 16 = -16\).
Remember, with negative bases, if the base itself is not in parentheses, apply the negative sign after doing the exponentiation.
First, calculate \(2^4\), which means multiplying \(2\) four times: \(2 \times 2 \times 2 \times 2 = 16\).
Next, apply the negative sign in front of the base after evaluating the exponent. Therefore, \(-2^4\) is computed as \(-1 \times 16 = -16\).
Remember, with negative bases, if the base itself is not in parentheses, apply the negative sign after doing the exponentiation.
Reciprocal
The reciprocal of a number is essentially "flipping" the number over. For a non-zero integer or a fraction, this involves exchanging the numerator and the denominator. When you see an expression like \(3^{-1}\), you're asked to find the reciprocal of \(3\), because the exponent \(-1\) indicates an inverse.
To find the reciprocal:
To find the reciprocal:
- Start with the number \(3\).
- Flip it to become \(\frac{1}{3}\).
Common Denominator
When adding or subtracting fractions, a common denominator is needed so the fractions can be combined properly. A common denominator is a shared multiple of the denominators involved. In our expression, we have two fractions: \(-\frac{16}{1}\) and \(\frac{1}{3}\).
To find a common denominator:
To find a common denominator:
- Look at the denominators \(1\) and \(3\).
- The least common denominator (LCD) is the smallest number that both \(1\) and \(3\) go into evenly, which is \(3\).
- Rewrite \(-\frac{16}{1}\) so it has this common denominator: Multiply both the numerator and the denominator by \(3\) to get \(-\frac{48}{3}\).
Integer Operations
Integer operations are basic arithmetic calculations including addition, subtraction, multiplication, and division performed on whole numbers. After converting expressions to fractions with a common denominator, integer operations are crucial to calculate the sum or difference. In our case, we need to perform the operation: \(-\frac{48}{3} + \frac{1}{3}\).
Here's how to handle the addition:
Here's how to handle the addition:
- Since the denominators are the same, directly add the numerators: \(-48 + 1\).
- This simplifies to \(-47\).
- Retain the common denominator, yielding the final result: \(-\frac{47}{3}\).
