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Chapter 2: Problem 23

Simplify \(24 \cdot 2^{-9}\) A. \(\frac{1}{32}\) B. \(\frac{1}{10}\) C. -10 D. -32

Short Answer

Expert verified
\(24 \cdot 2^{-9} = \frac{3}{64}\).

Step by step solution

01

Understand the Expression

The given expression is \(24 \cdot 2^{-9}\). The goal is to simplify this expression.
02

Rewrite the Exponential Term

Recall that a negative exponent means the reciprocal of the base raised to the positive exponent. Thus, \(2^{-9}\) can be rewritten as \(\frac{1}{2^9}\). So the expression becomes: \[ 24 \cdot \frac{1}{2^9} \]
03

Simplify the Multiplication

Simplify the multiplication by multiplying the numerator by 24: \[ \frac{24}{2^9} \]
04

Calculate the Denominator

Calculate the value of \(2^9\): \[ 2^9 = 512 \] So the expression now is \[\frac{24}{512}\]
05

Simplify the Fraction

Simplify the fraction by finding the greatest common divisor (GCD) of 24 and 512. The GCD is 8. Divide both the numerator and the denominator by 8: \[\frac{24 \div 8}{512 \div 8} =\frac{3}{64}\]

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Key Concepts

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

negative exponents
In mathematics, an exponent tells us how many times a number, known as the base, is multiplied by itself. When dealing with negative exponents, the concept might seem tricky but it's quite simple. A negative exponent indicates that we take the reciprocal of the base raised to the positive exponent. For example, when we have \( 2^{-3} \) , it can be rewritten as \( \frac{1}{2^3} \) . This means 2 is raised to the power of 3, but we take the reciprocal of the result. This concept can transform complicated-looking expressions into simpler fractions, making them easier to handle.
fraction simplification
Simplifying fractions is a fundamental skill in mathematics. It makes numbers easier to work with and understand. A fraction consists of a numerator (top part) and a denominator (bottom part). To simplify a fraction, you need to divide both the numerator and the denominator by their greatest common divisor (GCD). For example, in the fraction \( \frac{24}{512} \) , you can find the GCD of 24 and 512, which is 8. Dividing the numerator and denominator by 8 results in \( \frac{3}{64} \) . Simplified fractions are more neat and can make further calculations much easier.
greatest common divisor (GCD)
When simplifying fractions, finding the greatest common divisor (GCD) is essential. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. For instance, for the numbers 24 and 512, the GCD is 8. To find the GCD of two numbers:
  • List out the factors of each number.
  • Find the highest factor that appears in both lists.
Alternatively, you can use the Euclidean algorithm, which is a systematic method to find the GCD. Simplifying a fraction with the GCD ensures you get the simplest form of the fraction, making mathematical operations more straightforward.

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Most popular questions from this chapter

Write \(\sqrt{6}\) as an expression with an exponent. A. \(3^{\frac{1}{2}}\) B. \(3^1\) C. \(6^{\frac{1}{2}}\) D. \(\sqrt{6}^1\)

Simplify \(2^3 \cdot 4^3 \cdot 5^3\). A. \(40^9\) B. \(11^{27}\) C. \(40^3\) D. \(11^9\)

Simplify \(3 \cdot 3^3\). A. \(9^3\) B. \(3^4\) C. \(3^3\) D. \(9^4\)

Compute \(5^3\). Write your answer in the box.

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