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Chapter 0: Problem 27

Compute the numbers. $$(.01)^{-1.5}$$

Short Answer

Expert verified
1000

Step by step solution

01

Understand the exponentiation

The expression \((.01)^{-1.5}\) involves raising \(0.01\) to the power of \(-1.5\). A negative exponent indicates a reciprocal, and 1.5 as an exponent can be expressed as a square root and a power.
02

Rewrite the base in fractional form

Rewrite \(0.01\) as a fraction. \(0.01 = \frac{1}{100}\). So, the expression becomes \((\frac{1}{100})^{-1.5}\).
03

Apply the negative exponent rule

Recall that \((a/b)^{-n} = (b/a)^n\). Applying this rule, \((\frac{1}{100})^{-1.5} = (100/1)^{1.5} = 100^{1.5}\).
04

Simplify the exponent

Using properties of exponents, \(100^{1.5} = (100^1) \times (100^{0.5})\). \(100^1 = 100\) and \(100^{0.5} = \sqrt{100} = 10\).
05

Compute the final value

\(100 \times 10 = 1000\).

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Key Concepts

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

Negative Exponent
A negative exponent means you take the reciprocal of the base. For instance, if you have a number with an exponent of -1, you flip the base. If you have a bigger number, like -1.5, you still take the reciprocal and then apply the positive exponent.

For example, \(a^{-n} = \frac{1}{a^n}\). This means you turn the base upside down and change the sign of the exponent to positive.
Fraction to Power Conversion
Converting a decimal to a fraction helps make exponentiation easier. For example, turning 0.01 into \(\frac{1}{100}\) makes it simpler to apply rules of exponents.

This is very handy when you deal with roots or fractional exponents. When you have fractional bases, you should understand how to flip them properly using negative exponents. Here, \(0.01 = \frac{1}{100}\) and we applied the reciprocal rule to get it in an easier form to work with.
Properties of Exponents
Properties of exponents are tricks that help you solve problems faster. When raising a fraction to a negative exponent, you flip the fraction and switch the sign of the exponent.

Additionally, remember the property \((a^m)^n = a^{m \times n}\). This property allows you to break complex problems into manageable parts. For instance, \(100^{1.5} = 100^{1} \times 100^{0.5}\). Here, 100 raised to \(\frac{1}{2}\) is equal to the square root of 100.

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Most popular questions from this chapter

Let \(f(x)=x^{2} .\) Graph the functions \(f(x+1)\) \(f(x-1), f(x+2),\) and \(f(x-2) .\) Make a guess about the relationship between the graph of a general function \(f(x)\) and the graph of \(f(g(x)),\) where \(g(x)=x+a\) for some constant \(a\) Test your guess on the functions \(f(x)=x^{3}\) and \(f(x)=\sqrt{x}.\)

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$\frac{1}{y x^{-5}}$$

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$\frac{x^{-4}}{x^{3}}$$

Evaluate \(f(4)\). $$f(x)=x^{-1 / 2}$$

Evaluate \(f(4)\). $$f(x)=x^{0}$$
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