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Chapter 5: Problem 5

Simplify each expression. Write each result using positive exponents only. $$ \left(-\frac{1}{4}\right)^{-3} $$

Short Answer

Expert verified
The expression simplifies to -64.

Step by step solution

01

Understand the meaning of a negative exponent

Negative exponents indicate that the base is on the opposite side of a fraction. For instance, \[a^{-b} = \frac{1}{a^b}\]. Therefore, \((-\frac{1}{4})^{-3}\) means we need to take the reciprocal of \((-\frac{1}{4})\) to make the exponent positive.
02

Formulate the positive exponent

Taking the reciprocal of \(-\frac{1}{4}\), we get \(-4\). Hence, \[(-\frac{1}{4})^{-3} = (-4)^3.\]
03

Apply exponentiation to the new base

Now calculate \((-4)^3\), which means multiplying -4 by itself three times: \[(-4) \times (-4) \times (-4) = -64.\]
04

Finalize the expression with positive exponents

The expression \((-\frac{1}{4})^{-3}\) simplifies to \[-64\], as it already involves only positive exponents.

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Key Concepts

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

Negative Exponents
Negative exponents might seem a little tricky at first, but they have a simple meaning. When you see a negative exponent, it means you need to flip the base to the opposite side of a fraction. This process is done to get a positive exponent. For example,
  • If you have a number like \(a^{-b}\), it means that you are actually dealing with the reciprocal of the base raised to a positive exponent: \(\frac{1}{a^b}\).
This helps to rewrite expressions in a more straightforward way, usually involving positive exponents. Let's look at our math example:
  • \[\left(-\frac{1}{4}\right)^{-3} = \left(-4\right)^3\]
Reciprocal
The concept of a reciprocal is essential when dealing with negative exponents. A reciprocal is just flipping a fraction upside down. For a number expressed as a fraction like \(\frac{1}{4}\), its reciprocal would be \(4\). This is because the numerator and the denominator of the fraction swap places when finding the reciprocal.

For our specific example, we start with \(-\frac{1}{4}\). The reciprocal of \(-\frac{1}{4}\) is \(-4\), because again, you're flipping the fraction upside down. This step changes the negative exponent to a positive one, making it much easier to work with.
  • Reciprocal of \(-\frac{1}{4}\) is \(-4\)
  • Using this reciprocal turns \((-\frac{1}{4})^{-3}\) into \((-4)^3\)
Positive Exponents
After you've turned a negative exponent into a positive one, working with positive exponents becomes much easier and more straightforward. Positive exponents tell us how many times a number (base) is multiplied by itself.

Using the reciprocal process explained, the initial expression \((-\frac{1}{4})^{-3}\) becomes \((-4)^3\) with a positive exponent. Now you simply multiply the base \(-4\) by itself the number of times indicated by the exponent, which in this case is 3.
  • The calculation is as follows: \[(-4) \times (-4) \times (-4) = -64\]
So, positive exponents not only simplify mathematical expressions but also allow easy computation.
Simplification
Simplification is all about making an expression easier to understand and use, often by reducing it to its most basic form. In the context of exponents, this means expressing a number without unnecessary complexity, such as negative exponents or fractions that can be simplified.

In the previous steps, we turned \((-\frac{1}{4})^{-3}\) into \((-4)^3\), and then calculated it as \(-64\). There's nothing left to simplify because we have a single, whole number with a positive exponent.
  • Simplifying exponents often means turning negative exponents into positive ones.
  • It also involves reducing fractions or expressions to their simplest form.
  • The result should typically be a clearer, more straightforward expression.
In this example, we've followed the necessary steps to simplify completely. This achieves not only a clearer understanding but also an accurate solution!

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Most popular questions from this chapter

Simplify each expression by combining any like terms. $$ 7 w+w-2 w $$

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Multiply. $$ 3 x(2 x) $$

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The polynomial \(-20 x^{2}+156 x+14,437\) represents he electricity generated (in gigawatts) by geothermal sources in the United States during \(2002-2007\). The polynomial \(894 x^{2}-90 x+10,939\) represents the electricity generated (in gigawatts) by wind power in the United States during \(2002-2007\). In both polynomials, \(x\) represents the number of years after 2002 . Find a polynomial for the total electricity generated by both geothermal and wind power during 2002-2007. (Source: Based on information from the Energy Information Administration)
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