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Chapter 5: Problem 2

Simplify. Write each answer using positive exponents only. $$ \left(2^{-2}\right)^{2} $$

Short Answer

Expert verified
The simplified expression is \( \frac{1}{16} \).

Step by step solution

01

Apply the Power Rule

The first step in simplifying \( (2^{-2})^2 \) is to apply the power rule for exponents, which states that \( (a^m)^n = a^{m imes n} \). Here, \( a = 2^{-2} \), \( m = -2 \), and \( n = 2 \). Calculate \( -2 \times 2 \) to get \(-4\). Thus, \((2^{-2})^2 = 2^{-4}\).
02

Convert to Positive Exponents

Now, to express \( 2^{-4} \) with positive exponents, use the rule that \( a^{-b} = \frac{1}{a^b} \). Therefore, \( 2^{-4} = \frac{1}{2^4} \).
03

Simplify the Fraction

Calculate \( 2^4 \) by multiplying the base, 2, four times: \( 2 \times 2 \times 2 \times 2 = 16 \). So, \( \frac{1}{2^4} = \frac{1}{16} \).

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Key Concepts

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

Power Rule
The power rule is a fundamental principle for working with exponents. It states that when you have a base with an exponent raised to another power, such as \((a^m)^n\), you multiply the exponents together: \(a^{m \times n}\).
This rule simplifies expressions where a power is taken to another power. For example, in the expression \((2^{-2})^2\), applying the power rule helps us reduce it to a single exponent \(2^{-4}\).
Remember:
  • Multiply the exponents: \(-2 \times 2 = -4\).
  • The base remains unchanged.
This simplifies the expression and is crucial for further steps like converting negative exponents.
Exponent Rules
Exponent rules are essential for simplifying mathematical expressions and include handy rules for dealing with powers. Some key rules to remember include:
  • The power of a power rule: As seen earlier, \((a^m)^n = a^{m \times n}\).
  • A negative exponent \(a^{-b}\) can be converted into a positive one as \(\frac{1}{a^b}\).
  • Any number raised to the power of zero equals one: \(a^0 = 1\).

These rules make it easy to interpret and transform expressions. In the problem, the second rule helps convert \(2^{-4}\) into a fraction. This ensures all exponents are positive, making the expression easier to handle.
Simplifying Expressions
Simplifying expressions involves breaking down complex expressions into more manageable parts. This process often uses exponent rules and arithmetic operations. When simplifying, aim to:
  • Apply exponent rules effectively to reduce the expression to its simplest form.
  • Perform necessary arithmetic operations to calculate exact values.

In our example, the expression \((2^{-2})^2\) is simplified first using the power rule, then by converting exponents to positive values. Ultimately, it is reduced to \(\frac{1}{16}\). This not only makes the expression cleaner but also helps in understanding its value and potential use in equations.
Fractions
Fractions are a way of representing division and are crucial when dealing with negative exponents. When a number has a negative exponent, you can express it as a fraction. For instance, \(a^{-b}\) becomes \(\frac{1}{a^b}\).
This conversion:
  • Turns negative exponents into a more standard form that's easier to interpret.
  • Enables further simplification and evaluation, especially in calculations.

In the problem example, \(2^{-4}\) converts to the fraction \(\frac{1}{16}\), where \(2^4 = 16\). This conversion turns a potentially confusing negative exponent into a straightforward fraction.

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Most popular questions from this chapter

Factor. Assume that variables used as exponents represent positive integers. $$ x^{2 n}-36 $$

Find the value of \(c\) that makes each trinomial a perfect square trinomial. $$ n^{2}-2 n+c $$

Factor each polynomial completely. See Examples 1 through 12. $$ 8 x^{2}-26 x+15 $$

Factor each polynomial completely. See Examples 1 through 12. $$ 2(x+4)^{2}+3(x+4)-5 $$

Factor. Assume that variables used as exponents represent positive integers. $$ x^{4 n}-625 $$
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