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Chapter 5: Problem 1

Simplify each numerical expression. \(3^{-3}\)

Short Answer

Expert verified
\(3^{-3} = \frac{1}{27}\)

Step by step solution

01

Understand the Negative Exponent

A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For example, any number to the power of -1 is the reciprocal of the number. Hence, we can rewrite the expression \(3^{-3}\) as \(\frac{1}{3^3}\).
02

Compute the Exponential

Now, calculate \(3^3\). This is done by multiplying the base, 3, by itself repeatedly a total of three times: \(3 \times 3 \times 3 = 27\).
03

Simplify the Fraction

Replace \(3^3\) with 27 in the expression from Step 1 to complete the simplification. This gives us \(\frac{1}{27}\).

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Key Concepts

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

Reciprocal
When you encounter a negative exponent in mathematics, the key concept to understand is the reciprocal. The reciprocal of a number is simply flipping the number's numerator and denominator. For any given number or base \( a \), when raised to a negative exponent \( -n \), it means you take 1 over the base raised to the positive of that exponent: \( a^{-n} = \frac{1}{a^n} \).
  • In our example, \(3^{-3}\) means you take the reciprocal of three to the third power.
  • This transforms the expression into \( \frac{1}{3^3} \).
Understanding reciprocals helps simplify expressions with negative exponents by converting them to more familiar terms.
Exponential Computation
Once you've understood the concept of reciprocals and turned your expression into a positive exponent situation, the next step is the exponential computation. This involves calculating the power of a number. For \(3^3\), you would multiply the base, which is 3, by itself for the number of times indicated by the exponent, which here is 3. The computation follows as:
  • \(3 \times 3 = 9\)
  • Then \(9 \times 3 = 27\)
Thus, \(3^3 = 27\).
This computation allows you to express the power of a number in a simplified numerical form. This step is crucial before you move on to simplify the fraction in the final step.
Fraction Simplification
The final step in dealing with negative exponents once you've turned them into fractions involves simplifying the fraction if needed. In our scenario, once we computed \(3^3\) and found it to be 27, we write the expression \( \frac{1}{3^3} = \frac{1}{27} \).
  • This fraction is considered already simplified as 1 is in the numerator and 27 is not divisible by any other number except 1 and itself.
  • Simplifying fractions generally involves reducing them by finding the Greatest Common Divisor (GCD) between the numerator and denominator, although in this case, it's already fully reduced.
Simplifying fractions ensures the expression is as straightforward as possible, making it easier to communicate and understand in mathematical terms.

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Most popular questions from this chapter

Simplify each of the following. Express final results using positive exponents only. $$ \left(\frac{72 x^{\frac{3}{4}}}{6 x^{\frac{1}{2}}}\right)^{2} $$

For Problems \(1-18\), write each of the following in scientific notation. 812,000

Convert each number to scientific notation and perform the indicated operations. Express the result in ordinary decimal notation. $$ (0.000004)(120,000) $$

For Problems \(19-32\), write each of the following in ordinary decimal notation. $$ (9.14)\left(10^{-4}\right) $$

Simplify each of the following. Express final results using positive exponents only. $$ \left(\frac{x^{2}}{y^{3}}\right)^{-\frac{1}{2}} $$
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