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Chapter 0: Problem 66

Write each number in decimal notation without the use of exponents. $$9.2 \times 10^{2}$$

Short Answer

Expert verified
The decimal notation of \(9.2 \times 10^{2}\) is 920.

Step by step solution

01

Understand what the problem is asking

The problem is asking to rewrite the given value, \(9.2 \times 10^{2}\), in decimal notation. In this context, decimal notation means we wish to remove the exponent.
02

Apply the properties of exponents

\(9.2 \times 10^{2}\) means that we multiply 9.2 by 100 (since \(10^{2}\) equals 100). Since multiplying by 100 simply moves the decimal point two places to the right, we can quickly perform this step.
03

Write the final answer

So, by shifting the decimal two places to the right, \(9.2 \times 10^{2}\) becomes 920.

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Key Concepts

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

Understanding Exponents
Exponents are a way to express repeated multiplication of the same number. When you see an expression like \(10^2\), this is shorthand for \(10 \times 10\). The small number written to the upper right of a base number is called the "exponent" or "power."

Here鈥檚 what each part represents:
  • Base: This is the number that is being multiplied (in this case, 10).
  • Exponent: This indicates how many times the base is used as a factor (in this case, 2).
Let's break it down further:
  • \(10^2\) = \(10 \times 10 = 100\)
    • Understanding exponents is essential because they help us simplify operations and express large numbers succinctly. As you can see from the example above, rather than writing 100, you can simply use \(10^2\). Exponents play a crucial role in math and science, particularly when dealing with powers of ten.
    The Basics of Multiplication with Exponents
    When working with exponents, multiplication plays a significant role. If you encounter a problem involving a number with an exponent multiplication, you need to understand the underlying principles. One key scenario involves multiplying a number by 10 raised to an exponent.

    Imagine you have \(9.2 \times 10^2\), which is equivalent to multiplying 9.2 by 100. The key is understanding how the number is expanded:
    • The base number, 10, with an exponent of 2, becomes 100 because \(10 \times 10 = 100\).
    • Then you multiply 9.2 by 100, effectively shifting the decimal point two places to the right.
    The decisive factor here is that multiplication by powers of ten, like this, simply shifts the decimal point to the right, making it an efficient way to handle big numbers or decimal adjustments. It鈥檚 a tool that streamlines the process of scaling numbers up or down.
    Properties of Exponents
    When using exponents, it's beneficial to understand their properties as they simplify many mathematical calculations and expressions. Properties of exponents help you manage expressions with ease and fall into some common rules that dictate their behavior.

    For instance:
    • Multiplying like bases: \(a^m \times a^n = a^{m+n}\). If you multiply numbers with the same base, you add their exponents.
    • Zero exponent rule: Any base raised to the power of zero is 1, \(a^0 = 1\).
    • Power of a power: \((a^m)^n = a^{m \cdot n}\). When you raise an exponent to another exponent, you multiply the exponents together.
    Understanding these properties allows you to convert and manipulate expressions involving exponents efficiently. As seen in the original problem, knowing that \(10^2\) equals 100 is a direct application of these properties, making complex multiplications straightforward and manageable.

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    Most popular questions from this chapter

    In solving \(\sqrt{2 x-1}+2=x,\) why is it a good idea to isolate the radical term? What if we don't do this and simply square each side? Describe what happens.

    Your local electronics store is having an end-of-the-year sale. The price on a plasma television had been reduced by \(30 \%\) Now the sale price is reduced by another \(30 \% .\) If \(x\) is the television's original price, the sale price can be modeled by $$(x-0.3 x)-0.3(x-0.3 x)$$ a. Factor out \((x-0.3 x)\) from each term. Then simplify the resulting expression. b. Use the simplified expression from part (a) to answer these questions. With a \(30 \%\) reduction followed by a \(30 \%\) reduction, is the television selling at \(40 \%\) of its original price? If not, at what percentage of the original price is it selling?

    Solve each equation. $$\frac{1}{x^{2}-3 x+2}=\frac{1}{x+2}+\frac{5}{x^{2}-4}$$

    What does it mean to factor completely?

    How is the quadratic formula derived?
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