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Chapter 5: Problem 13

\(6^{-2}=\frac{1}{36}\)

Short Answer

Expert verified
The equation \(6^{-2} = \frac{1}{36}\) is confirmed to be correct after evaluating the negative exponent and simplifying the fraction.

Step by step solution

01

Understand Negative Exponent

Mathematically, a negative exponent represents a reciprocal. When a number 'a' is raised to the power '-n', it means 1 divided by that number 'a' raised to the power 'n'. Hence, \(6^{-2}\) can be rewritten to \(\frac{1}{{6^2}}\).
02

Simplify The Fraction

In this equation, \(6^2\) means '6' multiplied by itself, which equals to 36. Therefore, the expression could be simplified as \(\frac{1}{36}\).
03

Comparing The Two Sides

The left hand side of the equation is \(\frac{1}{36}\) and the right hand side is also \(\frac{1}{36}\). Since the two sides of the equation are equal, the original equation \(6^{-2} = \frac{1}{36}\) is correct.

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Key Concepts

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

Reciprocal
The idea of a reciprocal is tied closely to division and fractions. Imagine dividing 1 by a number, let's call it 'a'. The reciprocal of 'a' is simply \( \frac{1}{a} \). This means that instead of multiplying, you are essentially "flipping" the number to represent its opposite effect in multiplication.

Here are a few key points to understand about reciprocals:
  • Reciprocals multiply to give 1. For example, \( a \times \frac{1}{a} = 1 \).
  • A negative exponent like \( a^{-n} \) is saying "take the reciprocal of \( a^n \)." So, \( 6^{-2} \) becomes \( \frac{1}{6^2} \).
  • It's a handy way to deal with division in equations, allowing problems to be transformed and simplified.
Recognizing when to use reciprocals makes working with negative exponents much easier.
Simplification of Fractions
Simplification of fractions makes them easier to understand and work with. When you have a complex fraction, the goal is to break it down into its simplest form.

To simplify a fraction:
  • Find the greatest common divisor (GCD) of the numerator and the denominator. This involves finding the largest number that divides both the top and bottom numbers evenly.
  • Divide the numerator and the denominator by this GCD.
For our example, \( \frac{1}{36} \) is already in its simplest form because 1 is the numerator, and no number other than 1 divides 1 evenly. This fraction cannot be simplified further. Knowing how to identify when a fraction is fully reduced is crucial for solving math problems efficiently.
Powers of Numbers
Powers, or exponents, describe how many times a number is multiplied by itself. When you see an expression like \( 6^2 \), it means \( 6 \times 6 = 36 \). The small number above the base (also called the exponent) tells us how many times to use the base in a multiplication.

Some things to remember about powers are:
  • The base is the number being multiplied, and the exponent is how many times you multiply the base by itself.
  • Powers are a shortcut to signify repeated multiplication.
  • Negative exponents, such as \( 6^{-2} \), represent the reciprocal of the positive exponent: \( \frac{1}{6^2} \).
Understanding how powers work makes it possible to break down complex equations and solve them step-by-step.

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Most popular questions from this chapter

\(f(x)=\log _{3}(x-1)\)

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