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Chapter 5: Problem 23

For Problems \(1-30\), evaluate each numerical expression. $$ \frac{10^{2}}{10^{-1}} $$

Short Answer

Expert verified
The value of the expression is 1000.

Step by step solution

01

Evaluate the exponents

Simplify the expression by evaluating each exponent separately. Calculate \(10^2\) and \(10^{-1}\). \[10^2 = 100\]\[10^{-1} = \frac{1}{10} = 0.1\]
02

Set up the division

Place the values obtained from evaluating the exponents into the expression:\[\frac{100}{0.1}\]
03

Perform the division

Divide 100 by 0.1. Remember that dividing by a fraction (or decimal) is the same as multiplying by its reciprocal:\[\frac{100}{0.1} = 100 \times 10 = 1000\]
04

Verify the calculation

Double-check your calculations by conceptualizing the problem: dividing by 0.1 should move the decimal point one place to the right, effectively multiplying the number by 10. This confirms that: \[100 \times 10 = 1000\]

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Key Concepts

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

Exponents
Exponents are a mathematical way to express repeated multiplication of the same number. When you see an expression like \(10^{2}\), it means \(10\) is multiplied by itself. So, \(10^2 = 10 \times 10 = 100\).
Exponents can be both positive and negative. Positive exponents, like \(10^2\), indicate multiplication. Negative exponents, on the other hand, show division. For example, \(10^{-1}\) is equivalent to \(\frac{1}{10}\), because it means \(\frac{1}{10^1}\).
  • Positive exponent: Number is multiplied by itself.
  • Negative exponent: Number is inverted and represents division.
This concept helps simplify complex calculations, especially when dealing with powers of ten.
Division
Division is a fundamental arithmetic operation that represents splitting a number into equal parts. When you divide, you determine how many times a number (the divisor) fits into another number (the dividend).
In our problem, we perform \(\frac{100}{0.1}\). Division in mathematics sometimes requires us to work with decimals or fractions. The operation \(\frac{100}{0.1}\) can be seen as asking how many times \(0.1\) splices into \(100\).
  • Dividend: The number being divided (100).
  • Divisor: The number dividing the dividend (0.1).
Understanding division is crucial for solving problems where numbers need to be distributed or compared.
Decimals
Decimals are used to represent fractions in a base-10 system, making it easier to write and calculate with fractions. In our exercise problem, we have \(0.1\) as a decimal.
Decimals can have one or more digits after the point, which represent fractions of ten, hundred, thousand, and so on. For example, \(0.1\) is \(\frac{1}{10}\).
  • A single decimal digit represents tenths.
  • Multiplying or dividing by 10 shifts the decimal to the right or left.
In the problem, using decimals illustrates how a small number can divide into a larger number, helping us understand scale and place value.
Reciprocals
Reciprocals offer a way to perform operations such as division more intuitively, especially with fractions and decimals. The reciprocal of a number \(a\) is \(\frac{1}{a}\). For any nonzero number, multiplying it by its reciprocal equals one.
When dividing by a decimal like \(0.1\), you can multiply by its reciprocal, \(10\). This simplifies \(\frac{100}{0.1}\) to \(100 \times 10\).
  • A reciprocal flips a fraction.
  • Useful in division to avoid working directly with decimals.
This concept is crucial as it allows us to transform division into multiplication, which is often simpler to compute.

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Most popular questions from this chapter

Every whole number with a units digit of 5 can be represented by the expression \(10 x+5\), where \(x\) is a whole number. For example, \(35=10(3)+5\) and \(145=10(14)+5\). Now observe the following pattern for squaring such a number: $$ \begin{aligned} (10 x+5)^{2} &=100 x^{2}+100 x+25 \\ &=100 x(x+1)+25 \end{aligned} $$ The pattern inside the dashed box can be stated as "add 25 to the product of \(x, x+1\), and 100 ." Thus to mentally compute \(35^{2}\) we can think \(35^{2}=3(4)(100)+25=1225\). Compute each of the following numbers mentally and then check your answers. (a) \(15^{2}\) (b) \(25^{2}\) (c) \(45^{2}\) (d) \(55^{2}\) (e) \(65^{2}\) (f) \(75^{2}\) (g) \(85^{2}\) (h) \(95^{2}\) (i) \(105^{2}\)

Find the indicated products by using the shortcut pattern for multiplying binomials. $$ (-3 x+1)(9 x-2) $$

Find the indicated products by using the shortcut pattern for multiplying binomials. $$ (8 n+3)(9 n-4) $$

Find the indicated products by using the shortcut pattern for multiplying binomials. $$ (8 x-5)(3 x+7) $$

For Problems \(82-112\), use one of the appropriate patterns \((a+b)^{2}=a^{2}+2 a b+b^{2},(a-b)^{2}=a^{2}-2 a b+b^{2}\), or \((a+b)(a-b)=a^{2}-b^{2}\) to find the indicated products. $$ (x-4)^{2} $$
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