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Chapter 5: Problem 3

Simplify each numerical expression. \(-10^{-2}\)

Short Answer

Expert verified
The simplified form of \(-10^{-2}\) is \(-0.01\).

Step by step solution

01

Understand Negative Exponents

Negative exponents indicate the reciprocal of the base with the positive exponent. In other words, if you have a term with a negative exponent, like \( a^{-n} \), it can be rewritten as \( \frac{1}{a^n} \).
02

Convert Negative Exponent to Positive

Take the expression \(-10^{-2}\) and rewrite it using the concept of negative exponents. This becomes \(-\frac{1}{10^2}\) since the exponent \(-2\) becomes positive when moved to the denominator.
03

Calculate the Base Raised to the Power of 2

Now simplify \(10^2\). Calculate \(10 \times 10 = 100\). Thus, \(-\frac{1}{10^2}\) becomes \(-\frac{1}{100}\).
04

Simplify the Expression

The final step is understanding that the expression \(-\frac{1}{100}\) represents a negative fraction, and no further simplification is needed. The simplified version is \(-0.01\).

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Key Concepts

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

Exponentiation
Exponentiation is a way of expressing repeated multiplication of a number by itself. In our exercise, you encountered the term \(-10^{-2}\). Here, "10" is the base, and "-2" is the exponent. Negative exponents can seem tricky at first, but they have a straightforward interpretation.

The negative sign in an exponent means that instead of multiplying, we divide. Specifically, \(a^{-n}\) is equal to \(\frac{1}{a^n}\). This means we take the reciprocal of the base raised to the absolute value of the exponent. In simple terms, to deal with a negative exponent, flip the base to the denominator and convert the exponent to positive.
  • For example, \(10^{-2}\) becomes \(\frac{1}{10^2}\).
  • This property allows us to express equations without a negative exponent, making them easier to handle.
Fraction Simplification
Simplifying fractions involves rewriting them in their simplest form. In our solved problem, we converted a negative exponent to a positive fraction: \(-\frac{1}{10^2}\).

To simplify the fraction, you calculate the power of the base. For \(10^2\), multiply 10 by itself, which results in 100. The expression \(-\frac{1}{10^2}\) therefore simplifies to \(-\frac{1}{100}\). This fraction represents a very small negative number.
  • When simplifying fractions, ensure that the numerator and denominator are the smallest possible numbers representing the fraction.
  • Sometimes, simplification may involve dividing both the numerator and denominator by their greatest common divisor, but in our example, the fraction \(-\frac{1}{100}\) is already in its simplest form.
Basic Arithmetic Concepts
Understanding basic arithmetic concepts helps simplify expressions like \(-10^{-2}\). The core elements of arithmetic include addition, subtraction, multiplication, and division. In many cases, breaking down an expression into smaller, more manageable parts can lead to a simpler overall solution.

In this exercise, calculating \(10^2\) involved a basic multiplication operation: \(10\times10 = 100\). Knowing this multiplication step is crucial as it allows us to simplify other arithmetic expressions involving powers.
  • Basic arithmetic is not just about computation; it鈥檚 also about understanding the principles that guide each operation.
  • Becoming comfortable with these operations enables you to tackle more complex expressions, like those involving negative exponents, more intuitively.

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Most popular questions from this chapter

Simplify each of the following. Express final results using positive exponents only. $$ \left(\frac{2 x^{\frac{1}{3}}}{3 y^{\frac{1}{4}}}\right)^{4} $$

Use your calculator to estimate each of the following to the nearest one- thousandth. (a) \(7^{\frac{4}{3}}\) (b) \(10^{\frac{4}{3}}\) (c) \(12^{\frac{3}{5}}\) (d) \(19^{\frac{2}{3}}\) (e) \(7^{\frac{3}{4}}\) (f) \(10^{\frac{5}{4}}\)

Sometimes it is more convenient to express a number as a product of a power of 10 and a number that is not between 1 and 10 . For example, suppose that we want to calculate \(\sqrt{640,000}\). We can proceed as follows: \(\sqrt{640,000}=\sqrt{(64)\left(10^{4}\right)}\) $$ \begin{aligned} &=\left((64)\left(10^{4}\right)\right)^{\frac{1}{2}} \\ &=(64)^{\frac{1}{2}}\left(10^{4}\right)^{\frac{1}{2}} \\ &=(8)\left(10^{2}\right) \\ &=8(100)=800 \end{aligned} $$ Compute each of the following without a calculator, and then use a calculator to check your answers. (a) \(\sqrt{49,000,000}\) (b) \(\sqrt{0.0025}\) (c) \(\sqrt{14,400}\) (d) \(\sqrt{0.000121}\) (e) \(\sqrt[3]{27,000}\) (f) \(\sqrt[3]{0.000064}\)

For Problems \(1-18\), write each of the following in scientific notation. 117

Simplify each of the following. Express final results using positive exponents only. $$ \left(\frac{64 a^{\frac{1}{3}}}{16 a^{\frac{5}{9}}}\right)^{3} $$
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