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Chapter 5: Problem 2

For Problems \(1-30\), evaluate each numerical expression. $$ 2^{-5} $$

Short Answer

Expert verified
\(2^{-5} = \frac{1}{32}\).

Step by step solution

01

Understand Negative Exponents

Negative exponents represent the reciprocal of the base raised to the positive exponent. In this case, when you see the expression \(2^{-5}\), it means you need the reciprocal of \(2^5\).
02

Calculate the Reciprocal

Since \(2^{-5} = \frac{1}{2^5}\), you need to calculate \(2^5\) first. Calculate the reciprocal by recognizing that the expression equates to \(\frac{1}{32}\).
03

Compute the Exponent

Calculate \(2^5\). This involves multiplying the base, \(2\), by itself five times: \(2 \times 2 \times 2 \times 2 \times 2\). This results in \(32\).
04

Present the Final Answer

Since \(2^{-5} = \frac{1}{2^5}\), and we know \(2^5 = 32\), thus \(2^{-5} = \frac{1}{32}\).

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Key Concepts

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

Reciprocal
In mathematics, understanding the concept of a reciprocal is crucial when dealing with negative exponents. The reciprocal of a number is essentially its 'flip.' For a given number, the reciprocal can be expressed as one over that number.

For instance:
  • If you have the number \(2\), then its reciprocal is \(\frac{1}{2}\).
  • Similarly, for \(3\), the reciprocal is \(\frac{1}{3}\).
Knowing how to find a reciprocal becomes essential for interpreting expressions with negative exponents.

When you encounter a negative exponent, like in the expression \(2^{-5}\), you're being asked to find the reciprocal of \(2^5\). This is because the negative exponent signals that the base number should be flipped to its reciprocal and then followed by the positive exponent calculation.
Exponent Calculation
Exponents are a shorthand way of expressing repeated multiplication of the same number. When handling positive exponents, you multiply the base number by itself as many times as indicated by the exponent.

For example, \(2^5\) demands that we multiply \(2\) by itself five times. Here's how it is broken down:
  • Start with \(2\).
  • Multiply: \(2 \times 2 = 4\).
  • Next: \(4 \times 2 = 8\).
  • Continue: \(8 \times 2 = 16\).
  • Finally: \(16 \times 2 = 32\).
Thus, \(2^5 = 32\). The calculation of exponents is a fundamental part of evaluating expressions, especially when negative exponents call for calculating the reciprocal of these positive power values.
Numerical Expression Evaluation
When tasked with evaluating numerical expressions that involve negative exponents, it's important to methodically work through the problem to reach the correct answer.

Consider the expression \(2^{-5}\). Here's how you tackle it:
  • First, recognize the exponent is negative, meaning you look for a reciprocal: \(2^{-5} = \frac{1}{2^5}\).
  • Next, calculate the exponent as if it were positive: \(2^5 = 32\).
  • Finally, apply the reciprocal to complete the evaluation: \(\frac{1}{32}\).
Breaking down the problem using these steps ensures clarity. By understanding each part鈥攏egative exponents, reciprocal, and exponent calculation鈥攜ou can confidently evaluate the numerical expression and reach the accurate result.

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Most popular questions from this chapter

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. $$ (x+3)^{2} $$

Determine the number of terms in the product of \((x+y+z)\) and \((a+b+c)\) without doing the multiplication. Explain how you arrived at your answer.

Find the indicated products by using the shortcut pattern for multiplying binomials. $$ (6 n+5)(9 n-7) $$

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. $$ (1-5 n)^{2} $$

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. $$ (7 x-9 y)(7 x+9 y) $$
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