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Chapter 6: Problem 1

\(4^{3}=\)_______\(; 8^{2 / 3}=\)__________\(; 3^{-2}=\)________.

Short Answer

Expert verified
64; 4; \(\frac{1}{9}\)

Step by step solution

01

- Calculate \(4^{3}\)

First, we need to calculate \(4^{3}\). This means raising 4 to the power of 3. \[4^{3} = 4 \times 4 \times 4 = 64\]
02

- Calculate \(8^{2/3}\)

Next, we need to compute \(8^{2/3}\). This can be broken down as follows: First, find the cube root of 8: \[8^{1/3} = 2\] Then, raise this result to the power of 2: \[2^{2} = 4\] So, \(8^{2/3} = 4\).
03

- Calculate \(3^{-2}\)

Finally, we need to find the value of \(3^{-2}\). A negative exponent means taking the reciprocal: \[3^{-2} = \frac{1}{3^{2}} = \frac{1}{9}\]

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Key Concepts

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

raising powers
Raising a number to a power means multiplying that number by itself a specified number of times. For example, if you want to compute the value of \(4^3\), you would multiply 4 by itself two more times, like this:
\[4^3 = 4 \times 4 \times 4 = 64\]
Each time you multiply, that's one step in 'raising' the power. Some other key points when dealing with raising powers are:
  • The base is the number being multiplied.
  • The exponent is the number of times to multiply the base by itself.
  • Exponents can be positive, negative, or even fractions.
  • Common symbols for exponentiation include the caret (^), and in programming languages and calculators, you might see 'power' functions.
fractional exponents
Fractional exponents represent both roots and powers in one notation. For instance, if we consider \(8^{2/3}\), this can be understood as follows:
  • The denominator (3) tells you to take the cube root of the base (8).
  • Then, the numerator (2) tells you to square that result.
So, in the example:
First, find the cube root of 8: \[8^{1/3} = 2\]
Then, raise that result to the power of 2: \[2^2 = 4\]
Thus, \(8^{2/3} = 4\).
This multi-step process can be broken down for any similar problem by separating the fractional exponent into its root and power parts.
negative exponents
Negative exponents indicate a reciprocal. For example, if you have \(3^{-2}\), this means you take the reciprocal of \(3^2\). Following these steps, we get:
  • Compute \(3^2\): \[3^2 = 9\]
  • Take the reciprocal of that result: \[3^{-2} = \frac{1}{3^2} = \frac{1}{9}\]
Negative exponents can make large numbers small or fractional values smaller. It's helpful to remember that:
  • Any number to the power of 0 is 1: \(x^0 = 1\).
  • Using reciprocals can simplify complex expressions involving negative exponents.
Practicing with different bases and exponents will help you become comfortable with these concepts.

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Most popular questions from this chapter

Suppose that \(F(x)=\log _{2}(x+1)-3\) (a) What is the domain of \(F ?\) (b) What is \(F(7) ?\) What point is on the graph of \(F ?\) (c) If \(F(x)=-1,\) what is \(x ?\) What point is on the graph of \(F ?\) (d) What is the zero of \(F ?\)

Solve each equation. $$ 4 e^{x+1}=5 $$

Find the value of \(\log _{2} 2 \cdot \log _{2} 4 \cdot \log _{2} 8 \cdot \cdots \cdot \log _{2} 2^{n}\)

Use the Change-of-Base Formula and a calculator to evaluate each logarithm. Round your answer to three decimal places. \(\log _{5} 18\)

Solve each equation. $$ \log _{x} 16=2 $$
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