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Chapter 9: Problem 29

Divide as indicated. Write your answer using only positive exponents. $$\frac{2^{7}}{2^{9}}$$

Short Answer

Expert verified
The simplified expression is \( \frac{1}{4} \).

Step by step solution

01

Identify the Rule

When dividing exponential terms with the same base, we subtract the exponent in the denominator from the exponent in the numerator. This means that for the expression \( \frac{a^m}{a^n} \), it can be simplified as \( a^{m-n} \).
02

Apply the Exponent Rule

Apply the rule for division of exponents to \( \frac{2^7}{2^9} \). This results in \( 2^{7-9} \), simplifying to \( 2^{-2} \).
03

Convert to Positive Exponents

The negative exponent \( 2^{-2} \) can be converted into positive exponents by taking the reciprocal. Remember that \( a^{-n} = \frac{1}{a^n} \). So, \( 2^{-2} = \frac{1}{2^2} \).
04

Simplify the Expression

Calculate \( 2^2 \), which is equal to 4. Thus, \( \frac{1}{2^2} \) simplifies to \( \frac{1}{4} \).

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Key Concepts

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

Understanding Division of Exponents
When dividing exponents, especially those that share the same base, there is a straightforward rule you can apply. This rule says that you need to subtract the exponent in the denominator from the exponent in the numerator. In symbols, it looks like this: \( \frac{a^m}{a^n} = a^{m-n} \). This might seem tricky at first glance, but once you apply it to a problem, it makes things much simpler. Let's consider the expression \( \frac{2^7}{2^9} \).Here's how it works step-by-step:
  • Both the numerator and the denominator have the base of 2.
  • Subtract the exponent in the denominator (9) from the exponent in the numerator (7).
  • This gives you \( 2^{7-9} = 2^{-2} \).
By applying this rule, we've transformed the original fraction into a neat exponential expression with a negative exponent.
Explaining Negative Exponents
Negative exponents can initially seem a bit puzzling, but they're actually quite straightforward once you get the hang of them. The key to understanding negative exponents is to remember that they represent the reciprocal of the base raised to the positive equivalent of the exponent. In mathematical terms, \( a^{-n} = \frac{1}{a^n} \). Let's see this in action with our expression from before, \( 2^{-2} \):
  • Since \( 2^{-2} = \frac{1}{2^2} \), you're finding the reciprocal of \( 2^2 \).
  • This means instead of multiplying by 2, the multiplication turns into division.
  • Applying the rule simplifies the expression to \( \frac{1}{4} \).
Remembering this rule allows you to easily convert negative exponents into a form that's often simpler to work with in calculations.
Simplifying Expressions
Simplifying expressions involves breaking them down into their simplest form. When you simplify expressions with exponents, you're often looking to write the expression using only positive exponents. This can make the math easier to visualize and understand.Here's how you simplify \( \frac{1}{2^2} \):
  • The expression inside the denominator simplifies directly by calculating the power, which is \( 2^2 = 4 \).
  • Thus, \( \frac{1}{2^2} \) becomes \( \frac{1}{4} \).
Simplifying in this manner is a clear way to make expressions neater and often helps in recognizing patterns and easily solving related problems. Always aim for an expression with positive exponents to achieve a simplified and clear result.

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Most popular questions from this chapter

Multiply the following expressions. $$\left(x^{4}+x^{3}\right) x^{2}$$

Simplify the following expressions. \(\left(9 a^{5} b^{6}\right)^{2}\left(8 a^{5} b\right)\)

Simplify each expression, and write all answers in scientific notation. $$\frac{\left(3.5 \times 10^{-4}\right)\left(4.2 \times 10^{5}\right)}{7 \times 10^{3}}$$

Multiply the following expressions. \(\left(7 a^{3} b^{2}\right)\left(8 a b^{3}\right)\)

Find the value of each of the following expressions when \(x=4\). $$3 x-5$$
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