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Chapter 6: Problem 9

Cal and \(\mathrm{Al}\) 's teacher asked them, "What do you get when you square negative five?" Al said, "Negative five times negative five is positive twenty-five." Cal replied, "My calculator says negative twenty-five. Doesn't my calculator know how to do exponents?" Experiment with your calculator to see if you can find a way for Cal to get the correct answer.

Short Answer

Expert verified
Yes, Al is correct; the correct calculation should give 25. Use parentheses as \((-5)^2\) on calculators.

Step by step solution

01

Understand the Problem

Cal and Al need to determine the result of squaring negative five, which involves multiplying negative five by itself.
02

Squaring a Negative Number

Squaring a number means multiplying it by itself. So, \((-5)\times (-5)\) must be calculated.
03

Perform the Multiplication

Calculate \((-5)\times (-5)\). The product of two negative numbers is positive, so the result is \(+25\).
04

Analyze the Calculator Issue

Some calculators may interpret an expression like \(-5^2\) without parentheses as \(-(5^2)\), which results in \(-25\). To ensure the calculator squares the negative number correctly, the input should be \((-5)^2\).
05

Experiment with a Calculator

Enter \((-5)^2\) in the calculator to verify that it displays \(+25\). Avoid entering \(-5^2\) without parentheses.

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Key Concepts

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

Squaring Numbers
Squaring a number refers to multiplying the number by itself. When you square a number, you use it as a factor twice in multiplication. It is an important idea in mathematics because it frequently appears in algebra and geometry.

For instance, in the exercise given, squaring negative five means calculating \((-5) \times (-5)\). Since multiplying two negative numbers yields a positive result, the outcome is positive twenty-five. This concept is essential as it helps understand more complex equations and operations involving powers and polynomials. In general,\(a^2 = a \times a\), showing that the same number serves as both multipliers.
Calculator Use
Calculators are handy tools for arithmetic, but they have specific rules governing how they interpret numbers and operations. This can lead to confusion if you are not careful when entering expressions.

When calculating squares, especially of negative numbers, parentheses play a crucial role. For example:
  • If you input \((-5)^2\), the calculator understands you mean "negative five squared," giving the correct result of \(+25\).
  • However, typing \(-5^2\) without parentheses might be read as "negative of five squared," resulting in \(-25\).
To avoid such issues, always use parentheses when dealing with negative numbers in calculations. This way, the calculator processes the expression as intended, yielding the correct result every time.
Exponents
In mathematical terms, an exponent like 2 in \(5^2\) signifies using five as a factor two times, or multiplying five by itself. Exponents allow us to express repeated multiplication concisely. This is particularly useful when dealing with large numbers or powers.To compute a number with an exponent, you take the base 鈥� that's the number being multiplied 鈥� and multiply it by itself as many times as the exponent shows. For example, in \((-5)^2\):
  • The base is -5
  • The exponent is 2
  • This translates to \((-5) \times (-5)\), which simplifies to \(+25\) since the product of two negatives is positive.
Understanding and correctly using exponents is crucial for solving various mathematical problems ranging from basic algebra to advanced calculus.
Mathematical Notation
Mathematical notation is a system of symbols and signs used to represent numbers, operations, and concepts in a concise and standardized way. Correctly interpreting and using these notations is key to solving problems accurately.In the context of squaring negative numbers, mathematical notation clarifies the operation. Parentheses are particularly significant:
  • Without parentheses, writing \(-5^2\) implies taking the negative of the squared number, likely leading to incorrect solutions like \(-25\).
  • With parentheses, \((-5)^2\) explicitly directs the action of squaring both the base and its sign, ensuring the correct calculation \(+25\).
This correct use of notation ensures calculations align with mathematical rules, avoiding mistakes caused by incorrect interpretations. By paying attention to how problems are written, you ensure accuracy and clarity in your work.

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Most popular questions from this chapter

One of the most famous formulas in science is $$ E=m c^{2} $$ This equation, formulated by Albert Einstein in 1905 , describes the relationship between mass ( \(m\), measured in kilograms) and energy ( \(E\), measured in joules) and shows how they can be converted from one to the other. The variable \(c\) is the speed of light, \(3 \times 10^{8}\) meters per second. How much energy could be created from a 5 -kilogram bowling ball? Express your answer in scientific notation. James Joule (1818-1889) was one of the first scientists to study how energy was related to heat. At the time of his experiments, many scientists thought heat was a gas that seeped in and out of objects. The SI (metric) unit of energy was named in his honor.

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