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Chapter 4: Problem 16

Write each expression using a positive exponent. $$(-6)^{-2}$$

Short Answer

Expert verified
The expression (-6)^{-2} becomes \( \frac{1}{36} \) using a positive exponent.

Step by step solution

01

Understand Negative Exponents

A negative exponent indicates that the base should be taken as the reciprocal with a positive exponent. For example, if you have \((-a)^{-b}\), it can be rewritten as \(\frac{1}{(-a)^{b}}\). Our goal is to eliminate the negative exponent.
02

Rewrite as Reciprocal

Apply the rule of negative exponents to \((-6)^{-2}\). This means taking the reciprocal of \(-6\). So, we rewrite the expression as \(\frac{1}{(-6)^{2}}\).
03

Simplify the Expression

Simplify \((-6)^{2}\) by calculating the square of \(-6\). The square of \(-6\) is \(36\), because multiplying \(-6\) by itself gives \(36\).
04

Finalize the Expression

Substitute the simplified exponent back into the reciprocal form. Thus, \(\frac{1}{(-6)^{2}}\) simplifies to \(\frac{1}{36}\). This is the expression using a positive exponent.

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Key Concepts

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

Reciprocal and Negative Exponents
When dealing with negative exponents, one important concept to understand is the reciprocal. The reciprocal of a number is 1 divided by that number. When you have a negative exponent, like in the expression \((-6)^{-2}\), you're essentially being instructed to find the reciprocal of the base raised to the positive of that exponent.
This means transforming \((-6)^{-2}\) into \((1/(-6)^{2})\). This transformation simplifies the calculation and paves the path to turn the negative exponent into a positive one.
The reciprocal is a useful tool because it helps to simplify expressions involving negative exponents into more manageable forms.
Simplifying Exponents
Simplifying exponents is a key step when working with powers in mathematics. To simplify an expression containing exponents, you need to evaluate the power. In the example \((-6)^{-2}\), once the reciprocal is addressed, what remains is \((-6)^{2}\).

You then calculate the value of \((-6)^{2}\). Here is how you do it:
  • Multiply the base, \-6\, by itself.
  • So, \-6\ times \-6\ equals \36\.
Simplifying exponents by calculating these steps allows you to reduce complex expressions into simpler ones. This process is essential because it ensures that your expression is in its simplest and easiest to understand form.
Positive Exponents
Positive exponents are straightforward and signify repeated multiplication. Converting an exponent from negative to positive is often necessary for simplification. When an exponent is positive, it simply describes how many times to multiply the base by itself.
With \((-6)^{-2}\), after rewriting the expression as \((1/(-6)^{2})\) and simplifying it, you get a crisp expression with a positive exponent, specifically \(1/36\).

Here鈥檚 what positive exponents are all about:
  • They indicate repetitive multiplication.
  • They are easier to manage mathematically.
  • The outcome is conventionally positive when both the base and the result are handled accurately.
Understanding positive exponents helps in achieving cleaner, more manageable expressions during calculations.

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Most popular questions from this chapter

A nurse draws a sample of blood. A cubic millimeter of the blood contains \(22^{3}\) white blood cells and \(22^{5}\) red blood cells. Compare the number of white blood cells to the number of red blood cells as a fraction. \(\mathbf{F} \frac{10,648}{1}\) \(\mathrm{H} \frac{1}{484}\) \(\mathbf{G} \frac{484}{1} \quad \mathbf{J} \frac{1}{10,648}\)

Factor each monomial. $$22 a b^{3}$$

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. $$\frac{30}{37}$$

Find the greatest common factor of each set of numbers or monomials. $$18,28$$

Evaluate each expression if \(x=10, y=-5,\) and \(z=4 .\) Write as a fraction simplest form. $$\frac{1}{(z)(z)(z)}$$
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