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Chapter 4: Problem 15

Evaluate each expression. Do not use a calculator. $$-81^{0.5}$$

Short Answer

Expert verified
-9

Step by step solution

01

Identify the Expression

The expression given is \[-81^{0.5}\].This represents the calculation of -81 raised to the power of 0.5.
02

Understand the Meaning of the Exponent

The exponent 0.5 is equivalent to taking the square root, so \[81^{0.5} = \sqrt{81}\].This means we need to find the number that, when multiplied by itself, equals 81.
03

Calculate the Square Root of 81

Identify the number which squared (multiplied by itself) equals 81. That number is 9 because \[9\times 9 = 81\]. Therefore, \[\sqrt{81} = 9\].
04

Apply the Negative Sign

Since the original expression is negative, apply the negative sign obtained in the calculation, which makes the expression -9. Thus, \[-81^{0.5} = -9\].

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Key Concepts

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

Exponents
Exponents are a fundamental concept in mathematics where a number, known as the base, is raised to the power of another number, the exponent. The exponent tells you how many times to multiply the base by itself.
  • For example, in the expression \(5^3\), the base is 5, and the exponent is 3, indicating that 5 should be multiplied by itself three times: \(5 \times 5 \times 5 = 125\).
  • When the exponent is 1, the expression simplifies to the base itself. For example, \(a^1 = a\).
  • Any number raised to the power of 0 is always 1: \(a^0 = 1\), given that \(a eq 0\).
Exponents are used to simplify expressions and are essential for scientific notation and growth models. They are also important when working with polynomial and exponential functions.
Square Roots
The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, 9 is the square root of 81 because \(9 \times 9 = 81\).
  • The symbol for the square root is \(\sqrt{}\).
  • Square roots undo the operation of squaring a number. If \(b^2 = a\), then \(b = \sqrt{a}\).
  • Square roots can have both positive and negative results, but usually, the principal root (positive) is chosen unless specified otherwise.
In our exercise, the exponent 0.5 indicates a square root. To solve, we find that \(81^{0.5} = \sqrt{81} = 9\). In many cases, you may need to be cautious about domain restrictions when dealing with square roots, especially in different mathematical contexts.
Negative Numbers
Negative numbers are values less than zero, and they play an important role in mathematics. They are often encountered in financial calculations, temperature measurements, and physics.
  • Negative numbers are written with a minus sign, such as \(-3\) or \(-12\).
  • When multiplying or dividing numbers, a negative and a positive number yield a negative product or quotient: \(-a \times b = -(a \times b)\).
  • When both numbers are negative, their product or quotient is positive: \(-a \times -b = ab\).
In the expression \(-81^{0.5}\), the negative sign is applied after calculating the square root. Hence, the result is \(-9\). Understanding how negative numbers interact with other numbers and operations is essential for solving equations and interpreting mathematical expressions correctly.

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Most popular questions from this chapter

Consider the expression \(16^{-3 / 4}\) (a) Simplify this expression without using a calculator. Give the answer in both decimal and \(\frac{a}{b}\) form. (b) Write two different radical expressions that are equivalent to it, and use your calculator to evaluate them to show that the result is the same as the decimal form you found in part (a). (c) If your calculator has the capability to convert decimal numbers to fractions, use it to verify your results in part (a).

Solve each problem. The brightness or intensity of starlight varies inversely with the square of its distance from Earth. The Hubble Telescope can see stars whose intensities are \(\frac{1}{50}\) of the faintest star now seen by ground-based telescopes. Determine how much farther the Hubble Telescope can see into space than ground-based telescopes.

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For Exercises \(83-90\), the domains were determined in Exercises \(73-80 .\) Use a graph to (a) find the range, (b) give the largest open interval over which the function is increasing, (c) give the largest open interval over which the function is decreasing, and (d) solve the equation \(f(x)=0\) by observing the graph. $$f(x)=\sqrt[5]{x+32}$$

Solve each problem. Planetary Orbits The formula $$ f(x)=x^{1.5} $$ calculates the number of years it would take for a planet to orbit the sun if its average distance from the sun is \(x\) times farther than Earth. If there were a planet located 15 times farther from the sun than Earth, how many years would it take for the planet to orbit the sun?
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