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Chapter 5: Problem 2

Simplify each expression. Write each result using positive exponents only. $$ 6^{-2} $$

Short Answer

Expert verified
\(\frac{1}{36}\)

Step by step solution

01

Understand Negative Exponents

A negative exponent indicates that the base should be taken as the reciprocal raised to the opposite positive power. The expression \(x^{-n}\) is equivalent to \(\frac{1}{x^n}\).
02

Apply Negative Exponent Rule

Applying the rule for negative exponents to the expression \(6^{-2}\), we rewrite it as the reciprocal of \(6\) with a positive exponent: \(\frac{1}{6^2}\).
03

Calculate the Power

Now, calculate \(6^2\). \(6^2 = 6 \times 6 = 36\).
04

Write the Final Expression

Substituting the calculated power back into the reciprocal expression, we get \(\frac{1}{36}\). This is the simplified form using positive exponents.

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Key Concepts

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

Reciprocal
When dealing with negative exponents, the concept of a reciprocal becomes essential. A reciprocal of a number is found by swapping the numerator and the denominator. For example, the reciprocal of 5 is \(\frac{1}{5}\). In the context of negative exponents, this notion helps us convert negative exponents into positive ones.
  • The negative exponent indicates that you should "flip" the base, turning it into a fraction.
  • An expression like \(x^{-n}\) becomes \(\frac{1}{x^n}\) through the use of the reciprocal.
By understanding how to find the reciprocal, you can confidently tackle expressions involving negative exponents.
Positive Exponents
Positive exponents reflect how many times you multiply a base number by itself. Understanding how this works is essential for handling both simple and complex expressions.
  • If you see an expression like \(6^2\), it tells you to multiply 6 by itself, which equals 36.
  • In contrast to negative exponents, positive exponents denote straightforward multiplication without inversions.
Knowing how to work with positive exponents allows you to handle almost any algebraic expression with confidence. It simplifies tasks such as expanding expressions or finding powers of numbers.
Simplification of Algebraic Expressions
Simplifying algebraic expressions is a fundamental skill in mathematics. It makes complex expressions more understandable and easier to work with. The steps usually involve reducing or rewriting expressions while following specific mathematical rules.
  • Start by addressing any negative exponents by converting them into positive exponents using the reciprocal.
  • Calculate any positive exponent powers, simplifying the base raised to the exponent.
  • Combine like terms where possible to make the expression as simple as it can be.
In our example of simplifying \(6^{-2}\), turning it into \(\frac{1}{36}\) followed these principles, allowing us to express the original equation with positive exponents perfectly.

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Most popular questions from this chapter

Add or subtract as indicated. $$ (3 x-2+6 y)+(7 x-2-y) $$

$$ \left[\left(1.2 x^{2}-3 x+9.1\right)-\left(7.8 x^{2}-3.1+8\right)\right]+(1.2 x-6) $$

Subtract. $$ \left(-6 y^{2}+3 y-4\right)-\left(9 y^{2}-3 y\right) $$

Fill in the boxes so that each statement is true. (More than one answer may be possible for these exercises.) $$ 7^{\square}=\frac{1}{49} $$

Multiply. $$ 10 x^{2}\left(20 x y^{2}\right) $$
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