Problem 90
Question
\((-5)^{2}\)
Step-by-Step Solution
Verified Answer
The result of \((-5)^{2}\) is \(25\).
1Step 1: Identify the Base and the Exponent
In the expression \((-5)^{2}\), \(-5\) is the base and \(2\) is the exponent.
2Step 2: Understand the Operation
The exponent \(2\) indicates that the base \(-5\) should be multiplied by itself. This means we need to calculate \(-5 \times -5\).
3Step 3: Perform the Multiplication
Multiplying \(-5\) by itself results in \(25\) because \(-5 \times -5 = 25\).
Key Concepts
Base and exponentMultiplicationNegative numbers
Base and exponent
The terms 'base' and 'exponent' are fundamental in understanding powers and exponents. Let's break down what each term means, using the example \((-5)^2\):
- **Base**: The base is the number that is being multiplied. In our example, \(-5\) is the base.
- **Exponent**: The exponent tells us how many times to multiply the base by itself. In \((-5)^2\), the exponent is \2\, meaning we multiply \(-5\) by itself twice.
Putting it all together in the expression \((-5)^2\), we understand it as \(-5 \times -5\). Exponents help simplify and shorten repeated multiplication into a manageable format.
- **Base**: The base is the number that is being multiplied. In our example, \(-5\) is the base.
- **Exponent**: The exponent tells us how many times to multiply the base by itself. In \((-5)^2\), the exponent is \2\, meaning we multiply \(-5\) by itself twice.
Putting it all together in the expression \((-5)^2\), we understand it as \(-5 \times -5\). Exponents help simplify and shorten repeated multiplication into a manageable format.
Multiplication
Multiplication is one of the basic operations in mathematics, where we combine equal groups. Here’s a quick overview of multiplication:
1. **Equal Groups**: Think of multiplication as taking equal groups of a certain quantity. For instance, \3 \times 4\ means three groups of four.
2. **Multiplying Bases with Exponents**: When you have an exponent, you are essentially multiplying the base by itself multiple times. As we saw in the problem \((-5)^2\), the exponent \2\ means we multiply \(-5\) by itself: \(-5 \times -5\).
3. **Negative Times Negative**: It’s crucial to know that multiplying two negative numbers yields a positive result. Hence, \(-5 \times -5 = 25\) because:
1. **Equal Groups**: Think of multiplication as taking equal groups of a certain quantity. For instance, \3 \times 4\ means three groups of four.
2. **Multiplying Bases with Exponents**: When you have an exponent, you are essentially multiplying the base by itself multiple times. As we saw in the problem \((-5)^2\), the exponent \2\ means we multiply \(-5\) by itself: \(-5 \times -5\).
3. **Negative Times Negative**: It’s crucial to know that multiplying two negative numbers yields a positive result. Hence, \(-5 \times -5 = 25\) because:
- Negative times negative equals positive.
- Therefore, \(-5 \times -5 = +25\).
Negative numbers
Negative numbers are numbers less than zero. They are shown with a minus sign (\text{{-}}\). Here are some key points to remember about negative numbers, especially in multiplication:
- **Multiplying a Negative by a Positive**: The result is always negative. For example, \-5 \times 3 = -15\.
- **Multiplying Two Negatives**: The result is always positive. This is because the two negative signs cancel each other out, as demonstrated by \(-5)^{2}\ = \(-5 \times -5 = 25\).
- **Using Negatives with Exponents**: When you raise a negative number to an even exponent, the result is positive. But if the exponent is odd, the result is negative. For instance:
Remember, handling negative numbers correctly in operations is crucial for accurate results.
- **Multiplying a Negative by a Positive**: The result is always negative. For example, \-5 \times 3 = -15\.
- **Multiplying Two Negatives**: The result is always positive. This is because the two negative signs cancel each other out, as demonstrated by \(-5)^{2}\ = \(-5 \times -5 = 25\).
- **Using Negatives with Exponents**: When you raise a negative number to an even exponent, the result is positive. But if the exponent is odd, the result is negative. For instance:
- \text{(-2)^2 = 4}\text{ (positive because the exponent is even)}.
- \text{(-2)^3 = -8}\text{ (negative because the exponent is odd)}.
Remember, handling negative numbers correctly in operations is crucial for accurate results.
Other exercises in this chapter
Problem 89
\((-2)^{3}\)
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\(12+4^{2} \cdot 3 \div 2+4\)
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To add \(\frac{5}{6} x\) and \(\frac{1}{4} x\), the fractions can be rewritten with the common denominator 24 , or they can be rewritten with the least common d
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\(15+6^{2} \cdot 2 \div 3+9\)
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